Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
420 Category Seminar Proceedings Sydney Category Theory Seminar 1972/1 973
Edited by Gregory M. Kelly
SpringerVerlag Berlin.Heidelberg • New York 19 74
Prof. Dr. Gregory M. Kelly Department of Pure Mathematics University of Sydney New South Wales 2006 Australia
Library of Congress Cataloging in Publication Data
Category Seminar. Proceedings Sydney Category Seminar 1972/1973. (Lecture notes in mathematics ; 420) Includes bibliography and index. 1. Categories (Mathematics)Congresses. 2. Functor theoryCongresses. I. Kelly~ G. M., ed. II. Title. IIl. Series: Lecture notes in mathematics (Berlin) ; v. 420. QA3.L28 no. 420 [QA169] 510'.8s [512'.55] 7419483
AMS Subject Classifications (1970): 18 A15, 18 A35, 18 A40, 18 C15, 18D05, 18D15, 18D99, 18E35
ISBN 3540069666 SpringerVerlag Berlin • Heidelberg • New York ISBN 0387069666 SpringerVerlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by SpringerVerlag Berlin • Heidelberg 1974. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE
The 1972,
Sydney
Category
Theory
when the c a t e g o r y  t h e o r i s t s
conceived
the idea of m e e t i n g
collected
in this v o l u m e
has
emerged
represents although typing,
As editor take to turn
enough
were r e c e i v e d The papers
the c o m m o n are
theme
set theory,
in d i m e n s i o n
structure
is categories
monoidal monad, fields,
closed
and
If m o n o i d a l of c a t e g o r y papers
are
these
level.
His
since been which may
first adapted
the time
it would
delays
I owe a p o l o g ' first
with
than our local
structure.
category
two
a lot r i c h e r
structure; enriched
is the
because
of the
of sets with
and m o n o i d a l
categories,
categories,
categories
theorist
proximity;
If c a t e g o r i e s
theory
then the a n a l o g u e
categories
then B r i a n Day with two types
c l o s e d ones, two
about
by more
to the c a t e g o r y
closed
concerned
between
and the
what
bearing
his groups,
a rings,
are to the algebraist.
theory,
ially m o n o i d a l
the w r i t i n g  u p
to B r i a n Day, w h o s e
pure
although
with
categories,
and m o d u l e s
then
by i". But
so on, b e c o m e
done,
than a year ago.
to sets,
"increase
that
"1972/1973"
the w o r k was
for the u n f o r s e e n
is that of cat e$ories
of pure
The papers
of 1974.
sanguine
b e l o w are c o n n e c t e d
seen as a n a l o g o u s
analog u e
the m i d d l e
and e s p e c i a l l y
more
in Sydney
The t i t l e  d a t e
of it, along with
into theorems;
of
a large part of the m a t h e m a t i c s
I was e x c e s s i v e l y
ideas
in the m i d d l e
day each week.
the time at w h i c h
us past
ies to S p r i n g e r  V e r l a g papers
for a w h o l e
e i g h t e e n months.
the final details have brought
was born
at the three u n i v e r s i t i e s
represent
from its first accurately
Seminar
are
in some
is our
field
sense
theorist.
of structure
and a d j u n c t i o n s
 and with
all this m o r e o v e r
paper
a new a d j o i n t  f u n c t o r
by M i k k e l s e n
be seen as r e p l a c i n g
"enriched"
theorem,
topos
process
and espec
the r e l a t i o n s
at the
to the e l e m e n t a r y by a t w o  st e p
In fact his
 monoidal
structures; gives
the "fields"
which
has
situation,
and
the t r a n s f i n i t e
IV
tower
construction
to c o m p l e t i o n s earlier
and to m o n o i d a l
work on m o n o i d a l
on the r e f l e x i o n closed
of
structures
and gives
a great
the e m b e d d i n g great
of A p p l e g a t e
value
and Tierney; completions.
closed
monoidal
structures
closed
perceive
applications.
for coherence
closed
problems,
of t o p o l o g i c a l
the d e f i n i t i v e
My own papers As e q u a t i o n a l the c a t e g o r y
with
of sets
equational on the
structure
2category
In this
setting
the
general
category
from a knowledge
a subclass
of doctrines
I call
clubs,
or more
and a smaller
interplay
equational
gives
between
some
coherence
results
available
do well
to read
alone
are used
results
in the club
on any
2category.
is that
of finding
My first
concrete where
except
looks
representation
the coherence paper
level,
for
by
the
and my third
getting
the fuller
The reader
§i and
at
problem
concerns
case by specialization. last,
the d o c t r i n e
paper
and adjunctions;
at the d o c t r i n e
the first paper
of
generally
My second
structures
things
a doctrine,
subclass
as a w o r d  p r o b l e m .
species
on
called
a very
can be f o r m u l a t e d
to a m o n a d
also
of its algebras.
that admit
al~ebra.
if we want
with a given
to a 2monad,
problem"
cartesian
quickly
to u n i v e r s a l
 or on a more
"coherence
§i0,
would
which
in the last two papers.
The e l e m e n t a r y
2categorical
papers
and for Street's
Street
goes
is c o n t a i n e d
on in his papers
one sense his papers
background
needed
in a joint
to look more
are the most
of
theorems.
correspond
explicitly
what
at certain
species
of categories,
one,
to bicategories;
The reader will
in some sense
correspond
and
categories;
into a m o n o i d a l
to look deeply
 so c a t e g o r i e s
categories,
His third discusses
extensions
of his
his
to study m o n o i d a l
of any g i v e n
groups
combines
of functor
category
spaces.
elegance
correspond
algebras
like t o p o l o g i c a l
structures,
many
concrete
second
for functor
subcategories
of a n o n  m o n o i d a l
categories
His
on r e f l e c t i v e
and his fourth uses his techniques closed
along with a p p l i c a t i o n s
general;
deeply
both
for my
expository
paper.
at 2categories.
in another, the most
In
fundamental.
If the study of various
led to category theory,
that of structures borne by a category leads
i n e x o r a b l y to 2category theory. inside any 2category, ant.
Street's
representable
structures borne by a set has
Much of category theory can be done
and the arguments are then often more t r a n s p a r 
first paper looks at some things that can be done in a 2category  e s s e n t i a l l y the same thing as a finitely
2complete one, except that it need not contain a terminal object; which is seldom needed. at this level,
In p a r t i c u l a r he studies fibrations and b i f i b r a t i o n s
along with such things as p o i n t w i s e Kan extensions.
second paper, w h i c h uses the first,
investigates
2categories with a
structure so r i c h that we can imitate those arguments, Yoneda lemma, that depend upon the hom functor. 2category of categories,
His
i n c l u d i n g the
Even applied to the
it provides new proofs and thus contributes
the "elementary"
theory of categories;
ordered objects,
it throws new light on elementary topoi.
to
applied to the 2category of
The i n v e s t i g a t i o n s of Street and Kelly, at least ~, are to some degree tentative,
and they m e n t i o n m a n y o u t s t a n d i n g problems which may
be of interest to others. Because of the time the volume has been in preparation, believe
I
it a P p r o p r i a t e to give dates of r e c e p t i o n for the papers
(although I don't quite know what it means to "receive" my ~wn). order is that of the table of contents. 1973; Apr. Street:
1973; Feb.
1974; Feb.
July 1973; Feb.
1974.
Day:
Feb.
1974. KellyStreet: Kelly:
Nov.
The
1973 revised May Oct.
1973; Jan.
1973.
1974; May 1974.
G.M. Kelly 19 July 1974
Vl
TABLE OF CONTENTS
B r i a n Day
On a d j o i n t  f u n c t o r
factorization
i
On closed categories of functors II
2O
An e m b e d d i n g theorem for closed categories
55
Limit
65
spaces and closed span categories
G.M. Kelly and Ross Street
Review of the elements of 2categories
75
Ross Street
Fibrations
and Yoneda's lemma in a 2category
104
E l e m e n t a r y cosmoi
134
On clubs and doctrines
181
Doctrinal a d j u n c t i o n
257
G.M. Kelly
Coherence theorems
for lax algebras
and for d i s t r i b u t i v e
laws
281
ON A D J O I N T  F U N C T O R F A C T O R I S A T I O N by
This note contains an a l t e r n a t i v e a p p r o a c h to a result of Applegate and Tierney S~T:
C ~ B
(~2J
and t3]) which states that an a d j u n c t i o n
over a suitably complete category B can be f a c t o r e d
through the full s u b c a t e g o r y of B d e t e r m i n e d by the objects are "orthogonal" to all the m o r p h i s m s
in B which
inverted by the functor S: B ~ C.
It is o b s e r v e d that a slight s t r e n g t h e n i n g of the completeness hypothesis on B gives a simple proof of this result. The f a c t o r i s a t i o n of the given adjoint pair takes place in two stages,
the first of which is a w e l l  k n o w n epicmonic
the given a d j u n c t i o n unit.
f a c t o r i s a t i o n of
This produces a full reflective
subcategory
B' of B having the p r o p e r t y that the class of objects which are orthogonal to any given class of morphisms B' of S is r e f l e c t i v e
in B'.
inverted by the r e s t r i c t i o n to
The combined result contains a theorem
of Fakir fll] which a s s o c i a t e s to each m o n a d T on B, the idempotent monad which inverts the same morphisms as T. case, where
For the relative
Vbased
V is a complete symmetric m o n o i d a l closed category,
result is closely related to a theorem by W o l f f /19~
the
§5.6 using co
completeness h y p o t h e s e s on B. Some of the o b s e r v a t i o n s made here are implicit in 121 and 13. However,
the r e l a t i o n s h i p
of category c o m p l e t i o n to epicmonic
isation, and to r e l a t i v e categories of q u a s i  t o p o l o g i c a l not d i s c u s s e d in the r e l a t i v e this article, ation,
Vbased version 191 •
the concepts of category,
etc., are a s s u m e d to be r e l a t i v e
m o n o i d a l closed category
functor,
Thus,
factor
spaces, was throughout
natural t r a n s f o r m 
to a suitable symmetric
V; this category is assumed to be locally
small with respect to a given c a r t e s i a n closed category S of "small"
sets and set maps. The work r e l a t i n g closure" G.M.
of the c a t e g o r y
Kelly,
isations of the
of S~T:
notations
and are as given
and [2]
contexts
cartesian
jointly The
with
general
and the r e l a t i o n s h i p
C ~ B to a m o n o i d a l
in the early parts
§i
The p r e l i m i n a r y
§2
Factorisation
§3
Categories
§4
Examples;
§5
The f a c t o r i s a t i o n
closed
structure
on
BOP),
[i0].
of Madjunctions.
completions
cribles. and m o n o i d a l
system
FACTORISATION
m in a Vcategory
is monic in B if,
closure.
for left adjoints.
THE P R E L I M I N A R Y
B(B,m)
called a strong monic
of E i l e n b e r g  K e l l y
are s t a n d a r d
factorisation.
that a m o r p h i s m
if the m o r p h i s m
theorem
are as follows:
of relative
§i
e in
was done
and the r e p r e s e n t a t i o n
sectionheadings
Recall
in other
"minimal
in §4.
The basic
The
spaces
in view of [I]
given here are useful
factorisation
2.5 and to the
of t o p o l o g i c a l
but not p u b l i s h e d
B is d i s c u s s e d
monic
to Example
in
V for each B C B.
for each
~(e,,,l)
I
B(l,m)
B(e,l)
A monic
epic e in B (that
the square
B(l,m)
B is called monic
is,
in B
m is for each
3
is a p u l l b a c k easily
diagram
established.
any s t r o n g m o n i c
which
(~,~):
(E,M)
Thus fix
initial
S~T:
is epic
The category for at least
(a)
E = (all epics
(b)
E = (all strong
(E,M)
throughout.
factorisation Let
be two versions
system b e t w e e n
B' be the full
B E B for w h i c h
a morphism The proof
B is assumed one
to have
of the f o l l o w i n g
canonical two cases:
strong monics
in B)
in BY and M = (all monics
in B~
(E,M)
(a) and
could be taken (b)
subcategory
in the
sense
of B d e t e r m i n e d
but we shall to be any proper of El31
52.3.
by the objects
An object
B E B is in B' if and only
if there
B ~ TC in M. is clear.
Proposition Proof.
C and B and an a d j u n c t i o n
~B E M.
Proposition i~. exists
53) are
and if a composite
which may be compared
In fact,
I151
then so is g.
in B) and M = (all epics
(cf.
in B is a strong monic,
is an isomorphism~
data are categories
C ~ B.
will
any e q u a l i s e r
is a strong monic
 factorisation
there
The usual p r o p e r t i e s
For example,
fg of two m o r p h i s m s The
in V.
1.2
The
inclusion
The r e f l e c t i o n
B' c B has a left adjoint.
sends B E B to the image
of ~B;
let
nB B
......
~ TSB
B denote
the
isomorphism Thus,
for
gram is
factorisation because each
B'
it ~ B',
an isomorphism
of is
an epic
the in
nB with
top
V, a s
n~ E E a n d m ~ M.
in
arrow
C with in
required:
the
left
inverse
following
T h e n Sn~ i s eSB
" Sm.
pullback
dia
an
B(n',l)
B(~,B,)
.........
1
^ B(B,TSB')
B(B,B')
t
B(n',I)
~ B(B,TSB')
^
C(SB,SB')
Moreover,
.
. . . C(Sn',l)
~ C(SB,SB').
because S: B ~ C inverts
ion, there results
an adjoint
the unit ~' of the reflect
triangle: ^
B
=
B'
C This choice
of
category
process M. of
terminology, called
In
is
clearly
other
words,
B, a n d
is
a closure B'
category
an adjunction
such
has
operation
the
(E,M)factorisations
equivalent as
for
S'tT
to with
B". unit
given as
Following in
a substandard
M shall
be
an Madjunctiqn.
Ex~ple the category S: [A°P,s]
1.3. [A°P,s]
Let A be the category of finite sets, of all functors
"spanning"
from A °p to S, and let
~ S be "evaluatlonatsingleton".
category of "simplicial subsets).
complexes"
The category
to S.
B' is the
(sets equipped with certain finite
If M = {all monies
in B'} then B" remains
valent to B', but if M is changed to {all strong monies is equivalent
let B be
equi
in B'} then B"
§2 Under
additional
factorisation following is first
F A C T O R I S A T I O N OF M  A D J U N C T I O N S completeness
of §i reduces
the given a d J u n c t i o n
form of a d j o i n t  f u n c t o r established
a category
theorem
for o r d i n a r y
C is M  c o m p l e t e
that C has the f o l l o w i n g
hypotheses,
setbased
limits
to one
in w h i c h
is applicable. categories.
if M is a s u b c a t e g o r y
inverse
the a d j o i n t  f u n c t o r
The t h e o r e m We
of monics
and M contains
the
say that in C such
each monic
so
formed: (a)
equalisers
(b)
pullbacks
(c)
all i n t e r s e c t i o n s
A functor limits
of morphisms.
of Mmonics
(i.e.
inverse
of Mmonics
T: C ~ B is M  c o n t i n u o u s
Mimages).
with a common
if it preserves
these
codomain. inverse
in C. Theorem
T: C  ~ B exists that,
of pairs
2.1.
If C is an Mcomplete
has a left adjoint
a "bounding"
family
for each C E C and
if and only (BB:
then a functor
if T is M  c o n t i n u o u s
and there
B E B) of m o r p h i s m s
in B such
B ~ TCB;
f E B(B,TC),
category
there
exists
a commuting
square:
BB B
~ TC B
i TC
~ TD Tm
with m C M. Proof. and the perties.
family
If T has a left
adjoint
then T is clearly
(nB; B E B) of a d j u n c t i o n
Conversely,
a left adjoint
for some m o r p h i s m factors
uniquely
with m E M and
B B factors
~B: B ~ TSB. through
let
(p,q)
~B"
has the r e q u i r e d
S: B ~ C is c o n s t r u c t e d
taking h: SB ~ C B to be the i n t e r s e c t i o n n: M ~ C B such that
units
Mcontinuous
through
by
in C of all the M  s u b o b j e c t s
Tn.
Moreover,
Then
B B factors
each m o r p h i s m
To see this,
be the p u l l b a c k
pro
f E B(B,TC)
let Tg.~ B = Tm.f:
in C of
(gh,m).
as Th.~ B
B ~ TD,
BB
TC B
Tg
TC
TD
' Tm
Then q E M and n B factors definition
of SB.
TSB as T ( p q  1 ) ~ B . be the
equaliser
through
Tq so q is an i s o m o r p h i s m
Thus
f, w h i c h
factors
This
factorisation
in C of (p,rq).
so e is an i s o m o r p h i s m
through
is unique.
Tp,
by the
factors
If f = Tr.~B,
Then e C M and ~B factors
by the d e f i n i t i o n
of SB.
through
This
let e
through
completes
Te
the
proof. Remark. V, §6 and §8) from T h e o r e m exist
Several of Freyd's
2.1 under
of the adjoint
in C and are p r e s e r v e d
that
C is c o t e n s o r e d
the c o t e n s o r i n g ;
this
which
2.2
the i n c l u s i o n
if and only
if there
by T: C ~
version
hypothesis
(cf. [18]
Chapter
may be r e c o v e r e d that enough products
B.
provides
2.1 we shall
simply
and that T: C ~ B p r e s e r v e s
a Vadjunction
by [16§4.1.
are b a s e d on the following:
Let C be an Mcomplete
full s u b c a t e g o r y
C C B is Mcontinuous. exists
each B E B and
B C E M for each C E C. follows
Then
an e n d o f u n c t o r
~: i ~ S such that
This
theorems
of T h e o r e m
transformation
Proof.
statements
as a Vcategory
assumption
Our a p p l i c a t i o n s Theorem
functor
the a d d i t i o n a l
To o b t a i n a Vbased assume
standard
C is r e f l e c t i v e
in B
S: B ~ B and a natural
BB factors
from T h e o r e m
of B for
through
an object
2.1 and the fact
that,
of C for
for
each m o r p h i s m
f: B ~ C in
B, with c o d o m a i n
Tf.B B ~ ~c.f by the n a t u r a l i t y
of
8, and
C E C, we have
~C E M for all C C C by
hypothesis. For a g i v e n class denote
the
full
subcategory
the t e r m i n o l o g y Zorth0$0nal
of f13])
if B(s,B)
if B is Z  l e f t  c l o s e d Now suppose proper
Vcategory)
Let TS:
for all
(~,n):
2.~.
S~T:
(E,M)
An object
forming
E is
s E Z (that is,
C ~ B is an M  a d j u n c t i o n
on B and let E denote
B Z c B has
for a
the class
of
B Z is closed
(and is c o t e n s o r e d
as a
a left Madjoint.
under
limits
endofunctor
(and cotensoring)
on B.
Because
of T h e o r e m
(orthogonally)
closed
in V for all B E B E then of a subset
of Z onto a complete
of B w h i c h
B E B is called
in
TC E B Z
2.2 are
satisfied
is an Madjunction.
the closure
Corollary
(following
of 114).
C E C, and ~B E M, the conditions
is an i s o m o r p h i s m
are
by S.
B ~ B be the desired
The class
B, let B Z
in V for each
If B is Mcomplete
Clearly
and the r e f l e c t i o n
in B w h i c h
to Z.
is an i s o m o r p h i s m
and if Z c Z then
Proof. B.
of all objects
orthogonal
system
in B i n v e r t e d
~orollary
in a Vcategory
in the t e r m i n o l o g y
that
factorisation
morphisms
Z of m o r p h i s m s
2.4.
contain
B
f ~ ~.
of E reflects
sublattice. The
in the sense
class
Thus
of
of subclasses
2.3 implies:
of all full r e f l e c t i v e
as a full r e f l e c t i v e
if B(f,B)
The o p e r a t i o n
the class
Corollary
that
subcategory
subcategories forms
a complete
lattice. Example spaces
2.5.
and continuous
determined
by all the
functors
are u s u a l l y
conta±ns
Top
Let
Top
maps.
denote Let
subfunctors called
the category
B be the
subcategory
of r e p r e s e n t a b l e
"eribies").
as a full r e f l e c t i v e
full
Then
subcategory.
of all t o p o l o g i c a l of
functors
B is locally
ITop°P,s~
(these Ssmall
Furthermore,
~nd
B ~ B' if
we take S:
tTop°P,s]
M ~ (all monics
~ S to be e v a l u a t i o n at the onepoint space and
in B).
Let T: B ~ B be the monad d e t e r m i n e d by the
functor w h i c h evaluates each crible at the onepoint space. is the class of all bijections classes of bijections, topological
spaces,
§3
in B and, by i n v e r t i n g a p p r o p r i a t e
one obtains r e f l e c t i v e
subcategories of quasi
limit spaces, and related structures,
the "minimal extension of
Top"
Then
including
discussed in Eli.
CATEGORIES OF RELATIVE CRIBLES.
Suppose h e n c e f o r t h that the given symmetric m o n o i d a l closed category V ~ (V, ®, I,I ,...) is Scomplete and admits all intersections of Msubobjects,
where M is fixed as either the class of mon
ics in V or the class of strong monics in V. theses imply that
V has canonical
ponding class E of epics in
These completeness hypo
(E,M)factorisations
V(cf. /15
P r o p o s i t i o n 4.5).
Categories of relative cribles are a practical Madjunctions.
for the corres
source of
Given a category C, each functor M: A ~ C generates
the ordinary c a t e g o r y ~o of "Mcribles" An M  c r i b l e is a functor t r a n s f o r m a t i o n F ~ C(M,C)
or "Mpreatlases"
(2J
§2).
F: A °p ~ V for which there exists a natural each of whose components
is in M.
A
m o r p h i s m from F to G of Mcribles is a natural t r a n s f o r m a t i o n from F to G. We shall call the functor M: A ~ C extendable M) if the limit C(MF,C)
fAFA, C(MA,C)I
exists in V and has a r e p r e s e n t a t i o n
for each Mcrible F. P r o p o s i t i o n 3.1.
If M: A ~ C is extendable then A
enrichment to a Vcategory A and M: A ~ C is a left Proof. A(F,G)
(with respect to
~ ~ A
o
admits
Vadjoint functor.
For each pair of Mcribles F and G, define
IFA,GAI
(cf. 151
§4).
This limit exists in V by virtue of
the M  e m b e d d i n g
/A [ FA, GA]
fA [ i,m]~
[FA, C(MA,C)] ~ C(~F,C) A
with the f o l l o w i n g lemma. Lemma.
If mAB:
S(AB) ~ T(AB)
is a natural family of Mmonlcs
b e t w e e n two functors from A °p ® A to V then the end of S exists
in V
if the end of T exists. Proof.
As in [9] P r o p o s i t i o n 111.2.2,
the end fA S(AA)
c o n s t r u c t e d directly as the i n t e r s e c t i o n in fA T(AA)
is
of all the pull
back diagrams:
[ A
PA 
, S(AA)
T(AA)
= T(AA).
The components mAB are all r e q u i r e d to be monic the induced family of morphisms
fA S(AA)
~ S(AA)
in order that
should be natural in
A. The functor category A inherits equalisers, pullbacks Mmonics,
and i n t e r s e c t i o n s of Mmonics
from V.
However,
for A to be
c o t e n s o r e d we shall in general suppose that C is cotensored; because the pointwise
cotensor /X,F]
of
this is
of X E V with F 6 A is then an
Mcrible by virtue of the M  e m b e d d i n g
[~FI
[ l ~ m ] [X, C(M,C)] ~ C(M, [X,C]).
The category A also inherits
(E,M)factorisation
from V and, by Prop
osition i.i, the relative Yoneda a d j u n c t i o n M~T: C ~ A is an Madjunction.
In other words, F: A °p ~ V is an Mcrible if and only
if the a s s o c i a t e d natural t r a n s f o r m a t i o n F ~ C(M, MF) has components in M. If the functor M: A ~ C is Mfaithful
in the sense that each
10 component of the canonical t r a n s f o r m a t i o n A(A) ~ C(M,MA) then every r e p r e s e n t a b l e
is in M,
functor from A °p to V is an Merible.
Thus
there is a dense Y o n e d a e m b e d d i n g YA: A ~ A with respect to w h i c h plays the role of the funetor category [A,V]; however A is a welldefined
Vcategory,
even when A is large.
P r o p o s i t i o n 3.2. then so is M: A ~
If M: A ~ C is extendable and Mfaithful
C.
Proof.
To prove that M is extendable,
Mcr~ble with t r a n s f o r m a t i o n ~F: KF ~ C(MF,C) Mfaithful,
let K: ~op ~ V be an
in M.
Because M is
the category A contains all the r e p r e s e n t a b l e functors
from A °p to V.
Thus we can substitute F = A(,A)
in ~ and obtain an
Mmonic
K(A(,A))  C(M(A(,A)),C) ~ C(MA,C). This makes K(A(,A)), Mcrible
r e g a r d e d as a functor in A E A °p, into an
so there exists a r e p r e s e n t a t i o n C(M(K(A(,A))),)
Thus, on d e f i n i n g ~
~ f [K(A(,A)), A
= M(K(A(,))),
c(~,)
C(MA,)]
we obtain
~ fA[X(A(,A)), c(~,)l
fA[ fF~F ~ FA, C(MA,)[ by the r e p r e s e n t a t i o n theorem,
~[KF, fA[FA, C(MA,)]] = ] [KF, C(MF,)],
F
as r e q u i r e d for M to be extendable. Mfaithful,
To verify that M: A ~ C is
consider the following commutative
square:
11
MFG ]A [FA,GAI
..
:,....C(MF,MG)
[ l,m] A
fA [ FA,C(MA,C)] ~f [ Z,C(1,r~)]
fA[FA,C(~,~Q)]
A where
m: MG ~ C c o r r e s p o n d s to m: G ~ C(M,C)
MFG E M because
f [l,ml A
E
M.
under adjunction.
Then
This completes the proof.
While the process of forming M from M is clearly not a closure operation, forms a monad.
the p r e c e d i n g result leads us to ask w h e t h e r it
For a fixed category C, the Mfaithful e x t e n d a b l e
functors M: A ~ C may be r e g a r d e d as a category M(C)
in which a mor
phism from M: A ~ C to N: B ~ C consists of a functor ¢: A ~ B and a natural i s o m o r p h i s m N~ ~ M.
Each such m o r p h i s m induces a r e s t r i c t i o n
functor ¢*: B ~ A which maps G E B to G~ ~ A; this functor preserves limits but in general does not commute with the a u g m e n t a t i o n s M and into C. Example iant "endofunctor" isomorphism)
3.3.
The c o n s t r u c t i o n of M from M becomes a covar
(composition being p r e s e r v e d only to within an
on M(C) if we replace ¢* by its left adjoint ~.
V = S, the existence of ¢ follows from T h e o r e m 2.1. exists as a left
For a general V,
Vadjoint if C is cotensored relative to V.
r e s u l t i n g " p r e c o m p l e t i o n monad"
on
YA: A ~ A as its unit and Y~: ~ ~ A
The
M(C) has the Yoneda e m b e d d i n g as its multiplication.
In the case where M: A ~ C has a right adjoint, alent to the c a t e g o r y of all cribles of A. suitably complete
For
A is equiv
If, in addition,
A is
(see P r o p o s i t i o n 4.5) then the Yoneda e m b e d d i n g Y
has a left adjoint which serves as an algebra structure for A with respect to this m o n a d on M(C). Conversely,
if S: A ~ A is the structure functor for any
12
a l g e b r a of the p r e c o m p l e t i o n monad then S is left adJoint to Y: A ~ with a d j u n c t i o n unit ~ , transformation;
where A: ~A ~ Y~ is the canonical natural
this follows from T h e o r e m 2.2.
i n d i c a t e d to the author by Anders Kock
§4
EXAMPLES;
E x a m p l e 4.1.
(cf. [17]).
C O M P L E T I O N S AND M O N O I D A L CLOSURE
As in §3, the base category V is S  c o m p l e t e and
admits all intersections
of Msubobjects
Let M: A ~ C be an M  f a i t h f u l functor whose direct
The role of ~ was
for the given choice of M.
extendable functor and let K: K ~ A be a
limit colim K exists in A.
Then M preserves
colim K if and only if M inverts the canonical t r a n s f o r m a t i o n s: colim A(, Kk) ~ A(, k Because each r e p r e s e n t a b l e there exists a largest
colim Kk). k
functor from A °p to V is orthogonal
to s
(relative to M) full r e f l e c t i v e subcategory S
of ~ for which the Yoneda e m b e d d i n g A C A factors through As and preserves colim K. This is the basis of many c o m p l e t i o n processes. ular,
In partic
if M: A ~ C is a strongly c o g e n e r a t i n g and c o l i m i t  p r e s e r v i n g
extendable full e m b e d d i n g into a cotensored category C then the facto r i s a t i o n of §2 yields:
z
A
C, M
where E is the class of all m o r p h i s m s a dense,
strongly cogenerating,
inverted by M.
Thus one obtains
continuous and cocontinuous e m b e d d i n g
A ~ AZ and the functor A~ ~ C reflects isomorphisms.
This gives an
alternative proof of [9] T h e o r e m 111.3.2. Remark. functor
The process of e x t e n d i n g an Mfaithful extendable
M: A ~ C to M: A * C and then forming the category of
13
fractions of A with respect to the class of all m o r p h i s m s inverted by defines a m o n a d on the category M(C) functors over C. monad"
of M  f a i t h f u l extendable
This monad is a quotient of the " p r e c o m p l e t i o n
on M(C) d e s c r i b e d in Example 3.3. The following two examples concern the r e l a t i o n s h i p of the
f a c t o r i s a t i o n s of 51 and §2 to a given m o n o i d a l
structure on the cat
egory B, and to the q u e s t i o n of m o n o i d a l closure c o n s i d e r e d in monoidal
l o c a l i s a t i o n [ 8].
be symmetric
The m o n o i d a l
for n o t a t i o n a l
simplicity,
structure on B is a s s u m e d to however the results can be
e s t a b l i s h e d in the more general setting of b i c a t e g o r i e s of B $ n a b o u [4]), b i c l o s e d bicategories, We recall from [7] called a normal reflective
and their localisation.
that a full reflective subcate~ory
enrichment to a m o n o i d a l adjunction.
(in the sense
subcategory of B is
if the a d j u n c t i o n admits The existence of such an enrich
ment implies that the r e f l e c t i n g functor preserves tensor products when B is m o n o i d a l
closed,
and,
is equivalent to the subcategory being
closed under e x p o n e n t i a t i o n in B (by [ 7] T h e o r e m 1.2). Example has
4.2.
Let SiT: C ~ B be an a d j u n c t i o n in which
([,M)factorisations,
r e f l e c t i o n B~~B.
as in §i, and let P: B ~ B' denote the
If B has a m o n o i d a l structure then P is a m o n o i d a l
l o c a l i s a t i o n in the sense of [81 A,B E B.
if P(A @ n{) is an i s o m o r p h i s m for all
By P r o p o s i t i o n 1.2, P(A @ q~) is the unique m o r p h i s m m a k i n g
both the f o l l o w i n g diagrams commute:
A®B
, ~,,,
I®T] '
m
P(
[email protected])
,, T S ( A ® B )
P(
[email protected]')
[email protected] = n'
where
B
q' E E and m E M.
TS(
[email protected]') : TS(
[email protected])
P(A~PB) m
Thus an obvious
sufficient c o n d i t i o n for
P(
[email protected]') to be an i s o m o r p h i s m is that A @ ~ for all A,B E B.
E E and TS(A @ ~ )
C M
14 In the case where the m o n o i d a l structure on B is closed, the condition A ® n~ E E for all A,B E B is a u t o m a t i c a l l y Moreover,
if S~T:
satisfied.
C ~ B is a m o n o i d a l adjunction then S n e c e s s a r i l y
preserves tensor products,
so TS(A @ ~ )
is always an isomorphism.
Thus the category B' becomes a normal reflective
subcategory of B.
This example is related to the situation discussed in [8] where B = [A°P,v]
for a small m o n o i d a l category A over V, and B is
a s s i g n e d the convolution structure: F @ G = fAA'FA ~ GA'
@ A(, A @ A')
IF,G] = fA IFA, a(A~)1 If M: A ~ C is a functor into a cocomplete category C then the category A of Mcribles is a normal reflective subcategory of A°P,v] if C(M(
[email protected]),C) ition I.i).
is an Mcrible
for all A E A and C E C (by [8]
Propos
This coincides with the p r e c e d i n g situation if C is
m o n o i d a l closed and M preserves tensor products. Example 4.3
Suppose
(s,n): S~T:
C ~ B is an M  a d j u n c t i o n
in w h i c h B has equalisers,
pullbacks of Msubobjects,
sections of Msubobjects.
Then, by Corollary 2.3, B Z g B has a left
adjoint if Z is any class of morphisms (orthogonally)
and all inter
in B inverted by S.
If Z is
closed then the r e f l e c t i o n functor coincides with the
p r o j e c t i o n of B onto the category of fractions of B with respect to Z. Suppose that B has a m o n o i d a l closed structure and let Z ° = {s E Z;
[email protected] E Z for all B E B} denote the m o n o i d a l interior of Z with respect to this structure. P r o p o s i t i o n 4.4. by S~T:
If Z is a class of morphisms
in B inverted
C ~ B then BZO is a normal r e f l e c t i v e subcategory of B.
Proof.
The left adjoint of BZO c B exists by Corollary 2.3.
To verify that BzO = {C E B; B(s,C)
an i s o m o r p h i s m for all s E Z °} is
closed under e x p o n e n t i a t i o n in B, choose objects B E B and C C B Because
[email protected] E Z ° for each s E Z ° we have that B(
[email protected],C)
Z O"
~ B(s,[ B,C] ) is
15
an i s o m o r p h i s m
for all
s E Z o.
Thus
[B,C]
is o r t h o g o n a l
to Z °, as
required. It follows Z is closed. subclasses closed
Thus,
embedding
of A itself
such an adjoint
Msubobjects
is the
A(,
R~F)
adjoint
structure to M).
This
R and
A criterion
if A is closed
of Msubobjects,
is a corollary
satisfied
admits
a unit
it admits category
of limit
The
embedded
w h i c h all example
functor.
reflective spaces
into a r e f l e c t i o n
is i n v e r t e d
a monoidal
for the e x i s t e n c e
of
lie in
Here
under
extendable the f o r m a t i o n of
2.2.
The c o m p l e t e n e s s
T: ~ ~ ~ m a p p i n g
F to
components M.
is that where
A = Top and M: Top ~ S
Top is c a r t e s i a n closed and conThe
cartesian
closure
extension"
discussed
in [ i] and
embedding
into the
cartesian
closed
of Fischer [12].
THE F A C T O R I S A T I O N
general
unit
and if
in A.
by A and the functor
to M is the "minimal
a normal
§5
of B which
and i n t e r s e c t i o n
of T h e o r e m
n: i ~ T with
~ A(,RMF)
respect
whose
on ~ induces
Top as a full r e f l e c t i v e subcategory. with
of
of
and extendable
If M: A ~ C is an M  f a i t h f u l
adjoint
is the u n d e r l y i n g  s e t
Top
subcategory
if
following:
4.5.
The m o t i v a t i n g
tains
lattice
can be r e f l e c t i v e l y
in A, then A is r e f l e c t i v e
are
F ~ C(M,~IF)
a left
closed
pullbacks
Proof. hypotheses
on the class
to the
if M: A ~ C is Mfaithful
A*A has
a right
of equalisers,
subcategory
(relative
Proposition with
restricts
closed
closure".
by M then each m o n o i d a l
functor
operation
of Z, and each r e f l e c t i v e
In particular, the Y o n e d a
Z ° is o r t h o g o n a l l y
the m o n o i d a l  i n t e r i o r
B E as a r e f l e c t i v e
in a "monoidal
closure
that
of Z = {f; Sf isomorphism}
subclasses
contains
from this result
process
followed
S Y S T E M FOR LEFT ADJOINTS.
of f a c t o r i n g
an adjoint
pair of functors
by an i s o m o r p h i s m  r e f l e c t i n g
embedding
of
16
has a global interpretation.
We
consider the "category"
Adj
for w h i c h
an object is a category which is Mcomplete with respect to a suitable
(E,M)
(see §§i and 2) f a c t o r i s a t i o n syst~m leftadjoint
functor;
the left adjointso
on it and a m o r p h i s m is a
c o m p o s i t i o n of morphisms is Just c o m p o s i t i o n of
The class of r e f l e c t i o n functors is d e n o t e d by R
and the class of i s o m o r p h i s m  r e f l e c t i n g left adjoints is denoted by N. P r o p o s i t i o n ~.i.
(R,N)
To w i t h i n natural i s o m o r p h i s m of functors,
forms a f a c t o r i s a t i o n system on Proof.
We use several facts from [13]
r e f l e c t i o n is an "epimorphism" adjoint, b e i n g faithful, functors;
Adj. §2.3.
and every i s o m o r p h i s m  r e f l e c t i n g left
is a "monomorphism"
to within i s o m o r p h i s m of
thus the f a c t o r i s a t i o n will be "proper".
leftadjoint
Moreover,
functor on an M  c o m p l e t e category, where
proper f a c t o r i s a t i o n (by §i and §2).
Clearly every
(E,M)
every
is a
system, has a f a c t o r i s a t i o n of the r e q u i r e d form
Finally, we verify that if a diagram of left adjoints: S A
~
F
B
~P//
H
J
C
~ D M
with S ~ R and M E N,
commutes to within an i s o m o r p h i s m then there
exists a left adjoint P such that PS ~ F and MP ~ H. (~,B): F~G, M~N,
and H  ~ K
and M reflects isomorphisms, Q = SG.
be the adjunctions.
Let
(~,~): S4E,
Because HS ~ MF
F factors through S as P = FE.
Let
To verify that P~Q it suffices to verify that G factors
through E; that is, that the m o r p h i s m ~GC: GC ~ESGC is an i s o m o r p h i s m for all C E C.
But,
for all C C C,
FqGC: FGC ~ F E S G C is an i s o m o r p h i s m because M reflects isomorphisms and MF~Gc ~ HSqGc which is an i s o m o r p h i s m because Sq is an isomorphism.
Define
PC to be
17
the composite: PC
ESGC
GF(ESGC)
~,GC
~~GFGC
.
G(FnGc )i Then pC.~GC adjunction implies
= I by naturality (a,6):
nGc'Pc
F~G.
= i.
of 6 and the triangle
Because
axioms
E is a full embedding,
This completes
the proof.
for the
this
18
REFERENCES
[1]
Antoine, P., Extension minimale de la catSgorie des espaces topologiques, C.R. Acad. Sc. Paris, t, 262 (1966), 13891392.
[2]
Applegate, H. and Tierney, M., Categories with models, Seminar on Triples and Categorical Homology Theory, Lecture Notes 80 (Springer 1969), 156244.
[3]
Applegate, H. and Tierney, M., Iterated cotriples, Reports of the Midwest Category Seminar IV, Lecture Notes 137 (Springer 1970), 5699.
[4]
[email protected], J., Introduction to bicategories, Reports of the Midwest Category Seminar I, Lecture Notes 47 (Springer 1967), 177.
[5]
Day, B.J. and Kelly, G.M., Enriched functor categories, Reports of the Midwest Category Seminar III, Lecture Notes 106 (Springer 1969), 178191.
(6]
Day, B.J., On closed categories of functors, Reports of the Midwest category Seminar IV, Lecture Notes 137 (Springer 197o), 138.
[71
Day, B.J., A reflection theorem for closed categories, J. Pure and Applied Algebra, Vol. 2, No. i (1972), iii.
[8]
Day, B.J., Note on monoidal localisation, Bull. Austral. Math. Soc., Vol. 8 (1973), 116.
[9]
Dubuc, E.J., Kan extensions in enriched category theory, Lecture Notes 145 (Springer
[io]
1970).
Eilenberg, S. and Kelly G.M., Closed categories, in Proc. Conf. on Categorical Algebra, La Jolla, 1965 (Springer 1966), 421562.
[11]
Fakir, S., Monade idempotente
[email protected] & une monade, C.R. Acad. Sc. Paris, t. 270 (1970), 99101.
19 [12]
Fischer, H.R., Limesr~ume,
Math. Annalen, Bd. 137 (1959)
269303. [13]
Freyd, P. and Kelly, G.M., Categories of continuous
functors
I, J. Pure and Applied Algebra, Vol. 2, No. 3 (1972), 169191. [14]
Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory, SpringerVerlag,
[15]
Kelly, G.M., Monomorphisms, Austral.
[16]
Berlin, 1967.
epimorphisms,
and pullbacks, J.
Math. Soc., Vol. 9 (1969), 124142.
Kelly, G.M., Adjunction for enriched categories,
Reports of the
Midwest Category Seminar III, Lecture Notes 106 (Springer 1969), 166177. [17]
Kock, A., Monads for which structures are adjoint to units (to appear).
[18]
Maciane, S., Categories for the working mathematician, ~erlag, New York, Heidelberg,
[19]
Wolff, H., Flocalisations 1973, 405438.
Springer
Berlin, 1971.
and Fmonads, J. Algebra, Vol. 24,
ON C L O S E D C A T E G O R I E S OF F U N C T O R S II
*
by Brian
In many product basic
examples
is c o n s t r u c t e d
generating
free
tensor
formal
product
or, more
can be e x t e n d e d biclosed
of this
a l o n g a dense
types
of dense
Yoneda
functors there type
functor
embedding on
A.
corresponds
C C B.
adjunction tiation
of
to this
N: A °p ÷ C is a dense embedding
of
are c o m b i n e d
A °p in [A,S]
*The r e s e a r c h here r e p o r t e d R e s e a r c h Council.
E7].
[A,S]
structure
adjoint
structure under
followed
is the
setvalued
on EA,SI.
is, the left biclosed
the two basic
on a small
category The
A
second
to a full then the
all e x p o n e n 
of B. in the
f o l l o w i n g manner.
then it can be d e c o m p o s e d
was
A
a monoidal
The first of all
if C is closed
functors
functor
on a category
by c o m b i n i n g
structure
that
if and only
two results
a
when a m o n o i d a l
structure
in [4] and
If B has a m o n o i d a l
by the i n t e r n a l  h o m
from the
consider
N: A °p + C to produce
question
biclosed
functor;
is
C.
To each p r o m o n o i d a l a monoidal
on a
example
groups
We shall
by a s k i n g
a promonoidal functor
A simple
groups.
construction
the tensor
structure
of two a b e l i a n
A °p into the category
is m o n o i d a l
The
product
considered
is the r e f l e c t i o n
embedding
in the category.
on the category
We give an answer
categories
of a m o n o i d a l
of two free a b e l i a n
generally,
structure
biclosed
as an e x t e n s i o n
of the tensor
generalisation
structure
of m o n o i d a l
set of objects
the c o n s t r u c t i o n
Day
by a r e f l e c t i o n
supported
by a grant
from
If
into the Yoneda [A,S]
to
from the D a n i s h
21
C p r o v i d e d C is sufficiently complete w i t h respect to A.
However,
this overall c o m p l e t e n e s s h y p o t h e s i s on C is g e n e r a l l y u n n e c e s s a r y in order to produce a m o n o i d a l biclosed
structure on
case w h e n N itself is a r e f l e c t i o n and
C, as it is in the
A °p is m o n o i d a l biclosed.
Thus we answer the original q u e s t i o n by first c o n s i d e r i n g a c o m p l e t i o n C* of C and finding conditions under which this c o m p l e t i o n is m o n o i d a l biclosed. completed
The structure we obtain on C is simply the trace of the structure C*.
The p r o c e d u r e produces many known c o n s t r u c t i o n s of tensorproduct functors and i n t e r n a l  h o m functors. the tensor product of algebras closed category, by L i n t o n lution"
it produces
for a m o n o i d a l m o n a d on a m o n o i d a l bi
and gives a conceptual
[18] and Kock [16].
In particular,
e x p l a n a t i o n of c o n s t r u c t i o n s
It also produces the canonical
structure on a functor category
[A,B] w h e n A
"convo
is a p r o m o n o i d a l
category and B is a suitably complete m o n o i d a l b i c l o s e d category. One advantage of this a p p r o a c h is that the coherence of the structure p r o d u c e d on C follows from the coherence of C* which, turn,
follows from the coherence results already e s t a b l i s h e d in [4]
and E7].
Completions can also be used to provide a concept of "change
of Vuniverse"
in the case where all categorical algebra is based on a
fixed symmetric m o n o i d a l closed category V. " Vstructure" large)
in
on any category [A,B]
This enables us to put a
of Vfunctors
Vcategory A to a suitably e n r i c h e d category
from a (possibly B.
This,
in
turn, makes the relative c o m p l e t i o n process a v a i l a b l e for large categories
V
(and u l t i m a t e l y leads to a r e d u c t i o n in algebraic computa
ion) . The c o m p l e t i o n process is used in sections
5,b, and 7 to exam
ine m o n o i d a l b i c l o s e d structures on categories of functors from a promonoidal
category to a m o n o i d a l b i c l o s e d category.
discuss b i c l o s e d categories of c o n t i n u o u s the work of B a s t i a n i  E h r e s m a n n
[2].
In section 6 we
functors and relate this to
Finally,
section 8 contains a
22
proof of the r e p r e s e n t a t i o n theorem for monads. The u n e x p l a i n e d notations and t e r m i n o l o g y used in this article are standard, and familiarity with the r e p r e s e n t a t i o n t h e o r e m is assumed [7],
(cf.
[ii] §i).
This m a t e r i a l is a development of results in [4],
[8], and [9] and is based on a doctoral thesis by the author
and [6]).
([5]
The thesis was supervised by Professor G.M. Kelly at the
U n i v e r s i t y of New South Wales.
The author has also b e n e f i t e d from
several d i s c u s s i o n s with A. Kock and R. Street.
The s e c t i o n  h e a d i n g s are as follows:
§i
R e f l e c t i o n in closed functor categories.
§2
The c o m p l e t i o n process.
§3
M o n o i d a l closed completion.
§4
MonQidal monads.
§5
B i c l o s e d functor categories.
§6
B i c l o s e d categories of continuous functors.
§7
C o m p l e t i o n of functor categories.
§8
Denseness
presentations.
§i R E F L E C T I O N IN CLOSED FUNCTOR C A T E G O R I E S The formulas needed for m o n o i d a l b i c l o s e d functor categories and r e f l e c t i o n of m o n o i d a l biclosed and [7].
S denotes
V = (V,®,I,[,],...)
structures are r e c a l l e d from [4]
"the" cartesian closed category of small sets and is a fixed symmetric m o n o i d a l
with small limits and colimits.
closed basecategory
All concepts of categorical algebra
are h e n c e f o r t h assumed to be relative to this V unless o t h e r w i s e stipulated. An a d j u n c t i o n
(~,~): SiT:
C ÷ B is called a m o n o i d a l reflect
ion or normal r e f l e c t i o n if T is a full embedding and the a d j u n c t i o n
23 data admits monoidal biclosed
enrichment.
If B = (B,@,I,/,\,...)
category then S~T is monoidal
following pairs of conditions notation)
is a monoidal
if and only if one of the
is satisfied
(where T is omitted
from the
for all B,B' E B and C E C: n: C/B ~ S(C/B)
]
n: B\C ~ S(B\C) q\l: SBkC ~ B\C
1
l/n: C/SB ~ C/B
]
S(
[email protected]):
S(
[email protected]')
~ S(
[email protected]')
]
S(
[email protected]):
S(B'@B)
~ S(B'®SB)
I
~ S(SB®SB').
}
S(n®D): S(
[email protected]')
A subcategory A C B is called class of functors ~ ( A ,  ) ;
A E A} jointly reflects
r e f l e c t i o n theorem for monoidal 1.2)
strongly generating
biclosed
states that the above conditions
in B if the
isomorphisms.
categories
The
(cf. [7] Theorem
are equivalent
to either of the
pairs n: D/A ~ S(D/A)
]
n: AkD ~ S(i\D)
J
S(n~l):
S(B~A)
~ S(
[email protected])
1
S(l®n):
S(
[email protected])
m S(A®SB)
J
(I.i)
for all B E B, A C A, and D E D, where A strongly generates B and D strongly
cogenerates
C.
If B is a monoidal
biclosed
category and C c B
category we say that C is closed under e x p o n e n t i a t i o n C/B have isomorphs generally functors
in C for all B E B
a stronger
and C e C.
condition than requiring
\ and / have restrictions
to
is a full subin B if B\C and
Note that his is
that the internalhom
C.
Given a small category A, each functor F e [A,V] expansion: F ~
[A FA @ A(A,)
: A~V
has an
24
Thus
each m o n o i d a l
is isomorphic
biclosed
structure
to a structure
of the
on the functor
following
[email protected] =
lAB FA @ GB @ P(AB
G/F =
lAB [P(AB),[FA,GB]]
category
form:
)
(1.2)
(1.3) F\G =
f
AB
[ P(AB) ,[ FA,GB] ]
where P(AB) = A ( A ,  )
® A(B,)
P: A °p @ A °p @ A ~ V
is the
A monoidal
biclosed
I and a s s o c i a t i v i t y
~ A(i,)
structure
The e x t e n s i o n called
of p r o m o n o i d a l
on
categories,
x(F)
= [ =
AB
[A
B.
such a c o l l e c t i o n
structure
of A with
IA(A,B), [IA,
components
[email protected] whose
for A and
§3), where
"o" denotes
(A,P,I,~,k,0)
on A to
[A,V]
is
V and it is an internal
hom
[ P ( A B C ) , [FA % GB,HC]] ABC
small p r o m o n o i d a l
product
by P and
with:
[FA,GB]]
[I,FA]]
and I are r e g a r d e d
and O  d i m e n s i o n a l
is d e t e r m i n e d
A.
the c o n v o l u t i o n
[A,V](F,G)
([4]
of a p r o m o n o i d a l
P(FGH) = [
P, Hom,
axioms
We call
of @ on [A,V].
m P(ABX) op(XC)
: IXoP(AX) coherence
functor
isomorphisms:
p = 0A
a promonoidal
of those
[A~V]
on
~ A(A,)
composition.
tensor
functor"
: IXoP(XA)
profunctor
Two
The r e s u l t i n g
k = kA
suitable
where
structure
P(AX)oP(BCX)
satisfying
sometimes
"structure
and identity
= ~ABC:
[A,V].
in
,
as the
2dimensional,
of p r o m o n o i d a l categories
(P,Hom,I)
The usual
1dimensional,
structure.
A and
B have a p r o m o n o i d a l
are the r e s p e c t i v e
isomorphism
tensor
of categories:
products
25
[A~B,V] then
becomes
an
~ [A,IB,VJ! isomorphism
of
To combine convolution monoidal
biclosed
structure
tive embedding where egory.
monoidal
categories.
with reflection,
and let S~T:
C contains
The Yoneda embedding
biclosed
let [A,V]
C ~ [A~V]
have a
be a full reflec
P as a strongly c o g e n e r a t i n g
A°p~[A,V]
is strongly
subcat
generating,
thus
we have: Proposition
i.i.
The adjunction
S~T:
C ~ [A,V]
is monoidal
if and only if the functors G/A(A,)
= /B [P(AB),GB]
A(A,)\G
: ~[P(AB),GB
have isomorphs
in C for all A E A
Proof, B = [A,V].
and G E P.
This is a r e f o r m u l a t i o n
of Condition
(i.I) with
For example, G/A(A,)
= /B'B [P(B'B),[A(A,B'),GB]]
/B
[P(AB),GB]
by (1.3)
by the r e p r e s e n t a t i o n
theorem
applied to B' E A. Remark.
[email protected] The concept
of convolution
in the sense of
[email protected] [3].
can be extended to include If
A = {Axy ; x,y E 0bj A} is a Vbicategory, [A,V]~
is a biclosed
with homcategories
{[Axy,V] ; x,y E Obj A}
bicategory
This consideration,
by formulas
in turn,
The extension of the results straightforward The reflection
Axy , then
analogous
leads to the concept
(1.3).
of a Vprobicategory.
in this article to Vbicategories
exercise once this conceptual theorem for biclosed
of the "oneobject"
to (1.2) and
theorem above.
is a
framework is established.
bicategories
is an exact analogue
A form of this theorem has been
2B
introduced by J. Meisen [19] in the study of relations Remark on symmetry.
The concept of a symmetric p r o m o n o i d a l
category is defined in [4] §3. resultwithsymmetry
in categories.
This produces an (obvious) analogous
for each result verified in the sequel.
§2
THE C O M P L E T I O N PROCESS
For this section the base category V will be assumed to have all i n t e r s e c t i o n s of m o n o m o r p h i s m s as well as all small limits and colimits. To each small category C we will assign a c o m p l e t i o n C* which is taken to be the largest full reflective contains the r e p r e s e n t a b l e subcategory.
subcategory of [c°P,v] which
functors as a full strongly c o g e n e r a t i n g
This coincides with the leftadjoint
f a c t o r i s a t i o n of the
c o n j u g a t i o n functor
[c°P,v] through
a reflection
~ [C,V]°P; followed
F ~ [ [FC,C(C,)], C by an i s o m o r p h i s m  r e f l e c t i n g
embedding,
denoted: [ c°P,vJ
;
C*
[ C,V] op The r e s u l t i n g embedding of C into its c o m p l e t i o n will be denoted by: E ~ E(C):
C ~ C*.
The basic p r o p e r t i e s of C* can be verified by e x a m i n i n g the c o t r i p l e  t o w e r c o n s t r u c t i o n of C* (Appelgate and Tierney
[i] and Dubuc
[i0]) or by using a direct method of a d j o i n t  f u n c t o r f a c t o r i s a t i o n ([9], Example 4.1).
There is a closely related t h e o r e m in W o l f f [22],
§5.6. The c o m p l e t i o n C* can be d e s c r i b e d e x p l i c i t l y as the full subcategory of [c°P,v]
of functors F such that ~
[Sc,FC]
is an i s o m o r p h ~ m
27
whenever
s is a m o r p h i s m in [c°P,v]
inverted by conjugation,
C* is equivalent to the c a t e g o r y of Vfractions of [c°P,v]
As such,
with respect
to the class of m o r p h i s m s inverted by conjugation. Remark.
The assignment C ~ C* is c a n o n i c a l l y functorial
w i t h i n isomorphism)
(to
on functors S: C ~ B which have a right adjoint,
and the image S*: C* ~ B*, S*E = ES, has a right adjoint. P r o p o s i t i o n 2.1.
If N: A ~ C is a dense functor then
EN: A ~ C* is dense. This fact is established in [8] Corollary
3.2.
C* is equivalent to the full subcategory of [A°P,v] that ~ [ S A , F A ]
More precisely,
of functors F such
is an i s o m o r p h i s m w h e n e v e r s is a m o r p h i s m in [A°P,v]
inverted by the r e s t r i c t e d  c o n j u g a t i o n
[A°P'v]
~ [C'v]°P;
functor:
F ~ [A [FA,C(NA,)] .
Thus the r e f l e c t i o n from [A°P,v]
to C* at F E [A°P,v]
has the value:
A
EN(F) = I FA.ENA, and this is isomorphic to E(f A FA.NA) w h e n e v e r the r e f l e c t i o n ~ P A . N A of F exists in
C.
The second m a i n o b s e r v a t i o n of this section is that the completion process provides a "structural change of Vuniverse".
Consider
the case V = S and C = V and let S* be a larger c a r t e s i a n closed category of sets c o n t a i n i n g S and V, and any other categories that we want to regard as "small", as internal category objects. be the c o m p l e t i o n of V with respect to S *.
Let W = V*
The basic p r o p e r t i e s of V*
(verified in [83) are as follows: Property i. Because
V °p is a symmetric m o n o i d a l category,
the
functor category [ v°P,s *] is symmetric m o n o i d a l closed and this structure extends the original structure of
V.
closed, the reflective embedding W C [ v°P,s*]
Because
V is monoidal
is monoidal,
hence the
embedding E: V ~ W preserves tensor product and internal hom.
This
28
implies
that
the
symmetric
embedded
in
WCat
internal
hom. Property
and dense,
VProf
(cf.
into
between
them,
closed
tensor
E:
is strongly
V ~ W Thus
consisting
is embedded
bicategory
VCat
category
preserves
coends.
Introduction),
the b i c l o s e d
profunctor
The e m b e d d i n g
it preserves
[3],
Vprofunctors
and the embedding
2.
hence
monoidal
WProf
is fully
product
the biclosed
cogenerating bicategory
of Vcategories
(but not
fully
and
and
on morphisms)
in such a way as to preserve
composition.
Property
3.
completion
IA,VJ*
equivalent
to
Vpromonoidal essentially
Because
E:
with respect
[A,WI
to W of a functor
(as v e r i f i e d
structure
unique
V * W is strongly
in [8]
on a small
Wpromonoidal
§4).
Vcategory
structure
cogenerating, category
This
IA,VI
ensures
is
that
A corresponds
on A when
the
each
to an
A is r e g a r d e d
as
a Wcategory. Because sections
W has all S *  s m a l l
of m o n o m o r p h i s m s ,
be an a r b i t r a r y hypotheses
products filtered
and colimits
the given base monoidal
closed
and all
inter
category
V can be assumed
to
category
(with no completeness
on it).
Remark cartesian
symmetric
limits
2.2.
closed
If V is a c a r t e s i a n
and the r e f l e c t i o n
([7] C o r o l l a r y colimits
2.1).
closed
category
[v°P,s *]
from
then W is
to W preserves
Thus
finite
products
in any c a r t e s i a n
closed
category,
commute
finite
with
because
this
is
so in [ v°P,s *] .
§3
MONOIDAL
CLOSED
Let N: A °p ~ C be a dense let W be a c o m p l e t i o n S* of sets w h i c h Wpromonoidal
functor
of V with respect
contains S, V, A,
category.
COMPLETION
and
between
two
to a c a r t e s i a n C.
Vcategories closed
Let A = (A,P,I,...)
and
category be a
29
P r o p o s i t i o n 3.i.
The reflective e m b e d d i n g C* C [A,W]
is monoi
dal if and only if the functors
[P(AB), C(NB,C)] B
(3.1)
[P(AB), C(NB,C)] B have isomorphs in C* for all A E A and C c C. Proof.
This is a restatement of P r o p o s i t i o n 1.1, with the
strongly c o g e n e r a t i n g class 9 C C* being Remark.
{C(N,C);
C E C}.
The objects C E C in (3.1) could be r e s t r i c t e d
further to lie in any subcategory 9 C C for which strongly cogenerates
C*.
For example,
is the limit of some functor
{C(N,D);
D E 9}
if D c C and each object C E C
(depending on C) with object values in 9,
then the embedding 9 C C followed by E: C ~ C* strongly cogenerates We shall consider the ~pecial case where the functors have isomorphs in C, and let C(N,H(AC)) representations.
and C(N,K(AC))
The i n t e r n a l  h o m and t e n s o r  p r o d u c t
C*.
(3.1)
be their
operations on C*
then provide functors /: C ® C °p ~ C*, \ C °p @ C ~ C*, and
[email protected]: C @ C ~ C* w i t h values:
D/C = fA [C(NA,C), C(N,H(AD))] C\D = f
[C(NA,C),
(3.2)
C(N,K(AD))]
A and AB
C @D = [
(C(NA,C)
C(NB,D) ) Q(AB)
(3.3)
where A Q(XY)
= ~
P(XYA).ENA.
The identity object of C* is: I = S A IA.ENA. Remark.
Note that, by the c o n s t r u c t i o n of the m o n o i d a l bi
closed c o m p l e t i o n C*, the r e p r e s e n t i n g objects H(AD) are isomorphic
(3.4)
and K(AD)
in C
in C* to the exponentials ED/ENA and ENA\ED respectively.
30 §4
Let that
is,
T = (T,~,q)
a monad
where
( T , T , T °) w i t h r e s p e c t natural
MONADS
be a m o n o i d a l T:
monad
on a m o n o i d a l
B ~ B has a m o n o i d a l
to w h i c h
p: T 2 ~ T a n d
functor
q:
category
B;
structure
i ~ T are m o n o i d a l
transformations. Let
egory
MONOIDAL
B(T)
exists
denote
over
the
the b a s e
category category
B(T)(CD)
~
of T  a l g e b r a s V when
B(CD)
over
V has t h e
.... B ( [ , 1 )
~
B.
This
cat
equalisers
B(TC,D)
B(TC,TD) for
all
K(T)
Talgebras
has
(C,~)
and
The monoidal
axioms
a monoidal
structure; 0:
is d e f i n e d
on objects
(D,~) on
K(T)
(as
(T,~,q) the
constructed imply
that
tensorproduct
® K(T)
in the
[10]
or
Kleisli
[16]). category
functor
~ K(T)
as it is in B a n d o n m o r p h i s m
objects
by t h e
components :
A$
K(T) (B,C)
B(B,TC)
.......
        ~ B ( A®B, AOTC ) ~A 0 B(I,I)
K(T)(A®B,AOC)
B(
[email protected],T(AOC) )
and
K(T)(A,C)

B(A,TC)
~
$ B
B(A®B,TCOB)
®B where
I and
co~nutativity
p are
the
o f the
, ,
K(T)(A®B,COB)
~ B(ASB,T(COB)),
B(l,p) left
and right
triangles:
actions
o f B on T d e f i n e d
by
31
A @ TC n @ i
TA @ TC
$ ® 7
TA @ C
T(AeC). It
is easily
verified =
that z.TX.p
t h e i , n t e r c h a n g e law =
uTo1
holds and this is equivalent to @ being a Vbifunctor. If T is a symmetric m o n o i d a l m o n a d on a symmetric m o n o i d a l category B then the interchange law corresponds to the " c o m m u t a t i v i t y law" of Kock [15].
The r e s u l t i n g m o n o i d a l
structure on K(I)
is then
symmetric. Lemma 4.1.
If a T  a l g e b r a is e x p o n e n t i a b l e
exponents are e v a l u a t i o n w i s e Proof.
in B then the
Talgebras.
If (C,~) is a Talgebra and B is an object of B then
the exponents C/B and B\C have algebra structures defined as the exponential transforms of the morphisms: B @ T(B\C)
1
>,, T(B @ ( B \ C ) ) \ T
e TC
~ ~
C
/Te T(C/B)
@ B
 T((C/B)
@ B)
P where e denotes the r e s p e c t i v e e v a l u a t i o n transformations. axioms for these structures follow d i r e c t l y Thus,
from those for ~.
if all Talgebras are e x p o n e n t i a b l e in S then we obtain
adjoint actions of B °p on P r o p o s i t i o n 4.2.
B(T).
The c o m p l e t i o n B(T)* is m o n o i d a l biclosed if
each Talgebra is e x p o n e n t i a b l e in Proof.
T h e algebra
Let
~.
(C,~) be a Talgebra.
Then the adjunctions:
32
B(~B,C)
~ S[,C/B)
B(B~,C)
~ B(,B\C)
provide adjunctions: B(T)(F(
[email protected]),C) ~ B(T)(F,C/B) B(T)(F(
[email protected]),C) ~ B(T)(F,B\C) where F: B ~ B(T) is the freealgebra
functor.
,
If these isomorphisms
remain natural when F is extended to the dense comparison N: K(Y) ~B(Y) ion 3.1.
(see §8) then B(Y)* is monoidal
By Proposition
the adjunctions)
biclosed by Proposit
8,2, it suffices to show (working out one of
that the composites:
N(T(C/B) @ B)
are equal.
functor
N(~I):
N((C/B)
~ B)
On composing both composites
Te ~
TC
, ~ ~ C
with ~ and filling in the def
initions of @ and ~C/B' we obtain a commuting diagram pair: T(C/B)®B
[email protected] ~, T(T(C/B)®A)
I
T 2 (C/B) ®B
N(
[email protected])
T(C/B)@B
,~,
¢((C/B)~B)
iTe
I
{C/B
C
TC
where ~C/B is the adjointtransform the adjointtransform T(C/B)
N(
[email protected])
n~
of ~C/B"
,
Then the lefthand
side is
of T2(C/B)
~ E
T(C/B)
~
~
C/B
T~
and these composites This completes
are equal because ~C/B is an algebra
structure.
the proof.
Under the hypothesis
of Proposition
the trace of B(T)* exists on B(T). there are natural transformations
First,
4.2 we can determine when for each Talgebra
(C,~),
33
Te.~:
C/B
Te.p:
B\C
~ TC/TB m TB\TC,
corresponding to the transformations: (C/B)
~ TB
TB ® (B\C) Proposition 4.3.
k ~ T((C/B)
@ B)
Te
TC
P ~ T(B @ (B\C)) ~ T C .
The category B ( T )
is biclosed in B(T)* if and
only if the equalisers:  ~ B\C
B~C
[\C
~ TB\C
TB\ TC and
CtB
, C / B
C/[
~ C/TB
TC/TB exist in B for all Talgebras Proof.
B(T).
Because K(T) is monoidal,
The equalisers
B(T)* C [K(T)°P,w]
the unit object NI lies in
are precisely the ends (3.2) by Proposition 8.2.
Proposition 4.4.
algebras
(B,~) and (C,~).
If the reflective
embedding
is monoidal then the tensor product in B(I)* of two
(C,~) and (D,~) lies in B(T) if and only if the coequaliser of
the reflective pair: T(TC~TD)
T2(C~D)
T(C~D)
exists in B(T). Proof.
The pair has a common right inverse T ( n ~ ) ;
that is, it
is reflective.
Moreover,
the coequaliser in B(T)
is then the joint
c o e q u a l i s e r of the pairs: T(
[email protected])
Tk ~ T2(C®D) ~ TO
T(C®D)
T(
[email protected])
.
By P r o p o s i t i o n 8.2, this coequaliser is the iterated coend
(3.3);
C ® D = [AB(B(T)(NA,C) @ B(T)(NB,D))N(
[email protected]) in B(T). this
B e c a u s e E:
completes
the
B(T) ~ B ( T ) * p r e s e r v e s
and r e f l e c t s
colimits,
proof.
The p r e c e d i n g
propositions
provide
an a l t e r n a t i v e
earlier work by Kock [16] and Linton [18].
approach
to
The use of c o m p l e t i o n leads
to a significant r e d u c t i o n in coherence computations. Another result of Kock [17], T h e o r e m 2.6, may be e s t a b l i s h e d using the c o m p l e t i o n method.
Namely,
if the base c a t e g o r y V is cartes
ian closed and B is cartesian closed over V then the category B(Y) is c a r t e s i a n closed if the functor T: B ~ B
preserves finite products.
This follows i m m e d i a t e l y from the fact that K(T) has the c a r t e s i a n monoidal structure,
hence the c o m p l e t i o n B(Y)* c [K(T)°P,w]
is c a r t e s i a n
closed. As m e n t i o n e d earlier,
the Kleisli c a t e g o r y K(T) of a symmetric
m o n o i d a l monad T has a symmetric monoidal ted structure on B(Y)* is symmetric.
structure, hence the comple
For a nonsymmetric m o n o i d a l monad
on a symmetric m o n o i d a l closed category B the tensor product might exist on B(T) but have no symmetry. Exampl e .
Let V = S and let ~: ~ ~ ~
of the discrete category ~ egory
of finite integers into the simplical cat
4, both with the o r d i n a l  s u m m o n o i d a l
is given the c o n v o l u t i o n m o n o i d a l ed on [~°P,s]
be the m o n o i d a l inclusion
by the (monadic)
structure.
If
[N,S]
structure then the structure g e n e r a t 
adjoint pair
35
¢
i [¢°P,l] : [~ °P,s]
is isomorphic to the c o n v o l u t i o n m o n o i d a l structure extends the n o n  s y m m e t r i c
§5
,[ ~,S] structure on [b°P,s] .
o r d i n a l  s u m structure on
This
~.
BICLOSED FUNCTOR CATEGORIES
When A and B are two Vcategories the category [A,B]
of all
Vcategory if V is suitably compl
Vfunctors from A to B exists as a ete; that is, if the end: [A,BI (F,G) = f A
B(FA,GA)
exists in V for all F,G C [A,B]. Lemma 5.i.
If the category B admits the V  t e n s o r  c o p o w e r s
A(A,).B for all A E A and B E B then the Yoneda functor
[A,B];
N: A°
[email protected] ~
N(AB)
= A(A,)B,
is dense. Proof.
[
[[A,B](N(AB),F), [A,B](N(AB),G)] A,B [A,B [B(B,FA),B(B,GA)I by d e f i n i t i o n of N,
f
B(FA,GA)
A by the r e p r e s e n t a t i o n t h e o r e m applied to B ~ = [A,BI(F,G), Remark. T h e o r e m 1.33)
as required.
This result was e s t a b l i s h e d by F. Ulmer
for the case where
V is the symmetric m o n o i d a l closed
category Ab of a b e l i a n groups and group homomorphisms, tensor product of abelian groups. eral Yoneda functor N: A°P®B
~
(20],
with the usual
In fact, Ulmer considers a more gen[A,BI, N(AB)
= A(A,).MB,
36
where M: B ~
B is an Abdense functor.
In order to investigate the existence of a m o n o i d a l b i c l o s e d structure on
[A,B],
first take W to be the c o m p l e t i o n of V with respect
to a c a r t e s i a n closed category S* of sets which contains A, B, V, and S, as "small" category objects.
Note that the a s s u m p t i o n that
A,B
exists as a Vcategory can be avoided by the use of such a c o m p l e t i o n W.
Let A = (A,P,J,...)
be a W  p r o m o n o i d a l
structure on A and let
B be a m o n o i d a l b i c l o s e d category relative to structure on B provides a promonoidal
V.
T h e n the m o n o i d a l
structure on B °p and c o n s e q u e n t l y
A ® B °p has the t e n s o r  p r o d u c t of p r o m o n o i d a l structures given by the expressions:
P((X,B),(Y,C),(,)) I(X,B) P r o p o s i t i o n 5.2.
~ PIXY)
~ B(,B~C)
~ JX ~ B(B,I).
The c o m p l e t i o n
[A,BI*
of [A,B]
with respect
to W is monoidal biclosed if the functor S [P(XYA),B(,GA)] : B °p ~ W A is r e p r e s e n t a b l e Proof.
for all X,Y E A and G ~ [A,B] . The first functor of (3.1), with the object
(a,b) E A °p @ B m a r k i n g the variable position,
becomes:
f [ P( (a,b), (A,B), (X,C)) ,[ A,BI (N(XC),a)l XC
= #
[P(aAX) ~ B(C,b~B),/ B(A(X,Y).C,GY)] XC Y
by the d e f i n i t i o n s of N and A ~ B °p,
[ [P(aAX), X by t h e
representation
theorem
[ [P(aAX), X because
B is
biclosed,
B(b®B,GX)] applied
B(b,GX/B)]
Y @ A a n d C E B,
37
B(b,H(A,B,G)(a)) by the representability hypothesis,
[ B(A(a,X).b, X
H(A,B,G)(X))
by the representation theorem applied to X C A,
[A,B] (N(ab), H(A,B,G)) by definition of N,
=
as required.
A similar computation reduces the second end in (3.1) to
the required form. Henceforth we assume that the hypothesis of Proposition 5.2 is satisfied, with natural isomorphisms:
~ [P(XYA),B(,GA/B)]
~ B(,H(Y,B,G)(X))
[ [P(XYA),B(,B\GA)] A
a B(,K(X,B,G)(Y))
1
(5.1)
for chosen representations H and K. Proposition 5.3.
The category
[A,B] is biclosed in [A,B]* if
and only if the ends: [AH(A'FA'G)
}
f ~(A,~A,a) A
exist in
[A,B] for all F,G E [A,B]. Proof.
The internalhom values (3.2) can be reduced as foll
OWS:
G/F = [
[[A,B] (N(AC),F), [A,B] (N,H(A,C,G))] AC [ [ A,B] ( A ( A ,  ) . C , F ) ,
[ A, B] (N,H(A,C,G))]
AC by the definition of N,
f
[B(C,FA), [A,B] (N,H(A,C,G))] AC by the r e p r e s e n t a t i o n theorem a p p l i e d to A E A, [
[A,B] (N,H(A,FA,G)) A
38 by the r e p r e s e n t a t i o n
theorem applied to C c B,
[A,B] (N, ~AH(A,FA,G)) ~A H(A,FA,G) Similarly,
if and only if the end
exists in [A,BI .
we obtain:
F~G ~ [A,B] (N, and this completes
~ K(A,FA,G)), A
the proof.
Thus the resulting
internalhom
Q/~ = ~H(A,FA,a)
functors on [A,B]
}
have values:
(5.2)
F\O = ~ K(A,FA,G) A and, by the definitions natural
of H and K in Proposition
5.2, there exist
isomorphisms:
G?N(AB) ~ H(A,B,G) N(AB)\G ~ K ( A , B , G ) . The internalhom
[A,B]*
functors have an identity object
if the identity of
lies in [A,B] ; this identity has the value: AB
I = f
by
I(AB)EN(AB)
(3.4),
= ~AB(jA @ B(B,I)).(A(A,).B)  J.l in [A,B]*,
by the r e p r e s e n t a t i o n
theorem applied
to A E A and B E B. The tensor product
in [A,BI*
of two functors F,G e [A,B]
the value: F @ G = fXBYC(B(B,FX)
@ B(C,GY)).Q(X,B,Y,C)
by
fxY Q(X,FX,Y,aY) by the r e p r e s e n t a t i o n
theorem applied to B,C E B, where AB
Q(X,FX,Y,GY)
= f
(P(XYA) ~ B(B,
[email protected]))EN(AB)
by d e f i n i t i o n of the promonoidal
category A @ B °p,
AB = ~
(P(XYA)
@ B(B,
[email protected])).E(A(A,).B)
(3.3),
has
39 by d e f i n i t i o n of N,
[AB(p(XYA) @ B ( B , F X ® G Y ) )  ( A ( A ,  ) . E B ) because E preserves tensoring,
P(XY).(
[email protected]) by t h e r e p r e s e n t a t i o n
theorem applied
t o A E A and B E B.
This establishes the following: P r o p o s i t i o n 5.4. r e s t r i c t i o n to [A,B]
The m o n o i d a l
structure on [A,B]* admits a
if the coend
F ~ G = [XY P(XY).(
[email protected]) exists in [A,B]
(5.3)
for all F,G E [A,B], and the identity object J.l of
[A,B]* lies in [A,B] . The coherence of the m o n o i d a l and biclosed structures on [A,B]
induced
is a consequence of the coherence of the m o n o i d a l biclosed
structure on [A,B]*.
In the case where B = V, the formulas
(5.3) reproduce the original c o n v o l u t i o n structure
(1.3) and
(5.2) and (1.2) on
[A,V]. Remark. A
It is s t r a i g h t f o r w a r d to verify,
using
(5.3), that if
and A' are two small promonoidal categories and if B is a sufficien
tly complete and cocomplete m o n o i d a l b i c l o s e d category then the canonical i s o m o r p h i s m of categories:
[A @ A',B]
~ [A,[A',B]]
admits enrichment to an isomorphism of m o n o i d a l b i c l o s e d categories, where A @ A' has the t e n s o r  p r o d u c t p r o m o n o i d a l
structure and each
funetor category has the "convolution" m o n o i d a l b i c l o s e d structure defined by (5.2) a n d (5.3).
4O
§6
BICLOSED
CATEGORIES
If A is a p r o m o n o i d a l egory
for w h i c h
functor
category
of [A,BI ctors.
the
subcategories
is assumed
to be
C.
For each
that
full
in [A,V] .
on the
subcategory
internalhom categories
of
for those
full
functors
Again,
fun
from
the base
cat
category
C let C m denote
ortho~onal
the class
of all m o r p h i s m s
the full
subcategory
of C oonsis
to Z (following
terminology
of
[12]);
is,
Let
~ denote
is an i s o m o r p h i s m
the ortho~onal
= {s E cm;c(s,C)
in V for all s E Z}.
c l o s u r e of Z in cm;
is an i s o m o r p h i s m
that
is,
in V for all C E CZ}.
C Z = C U = C~ for all Z C U c ~. For a given m o n o i d a l
Z c C m is called m o n o i d a l s @ C E Z; that both
by these
exist
cat
V.
C Z = (C E C;C(s,C)
Then
a given
this q u e s t i o n
Z of m o r p h i s m s
biclosed
(5.2)
of the Z  c o n t i n u o u s
Z C C m let C Z denote
ting of the objects
by
[2] on closed
us to examine
which consist
class
For a given
defined
all e x p o n e n t i a t i o n
leads
of [A,B]
A to B for a given
in
functors
The work of B a s t i a n i  E h r e s m a n n structures
FUNCTORS
and B is a m o n o i d a l
then we can ask w h e t h e r
is closed under
sketched
egory
category
internalhom
A,B]
OF C O N T I N U O U S
is,
structure
if C E C and
on the c a t e g o r y s E Z imply
if Z is stable under
C, a class
that
the m o n o i d a l
C @ s E Z and
action
of C on
sides of Cm. Remark.
with respect respect
It was
established
in [B]
§i that
to C then the category
C(Z I)
of Vfractions
to Z (as c o n s t r u c t e d
such that
each m o n o i d a l
in W o l f f
functor
has a unique m o n o i d a l  f u n c t o r
[21])
on C w h i c h
if Z is m o n o i d a l of C with
has a m o n o i d a l inverts
factorisation
through
structure
the elements
of Z
the p r o j e c t i o n
C ~ C(ZI). Proposition a monoidal
biclosed
6.1.
If A is a strongly
category
generating
subcategory
C and Z C C m then the f o l l o w i n g
are
of
41
equivalent: (a)
A E A and
(b)
C Z is c l o s e d
(e)
Thus
are
C/s and
isomorphisms lie
in
C.
If B C C
C(s,B\C)
are
and
C(
[email protected],C)
~ C(A,C/s)
for all A ~ A and
for all
B ~ C and
s E Z and
~ C(B,C/s)
s E Z, w h i c h
and
s E Z by
(a).
C E C Z because
and
C(s,C/B)
implies
that
A
m C(~s\C)
B\C
and
C E C Z.
s E ~ then
isomorphisms
(a) b e c a u s e
T h e n we can
[A,B] Z C [A,B]
[A,V~z
"Zcontinuous" all
Hence
C(
[email protected],C)
for all
~ C(s,B\C)
C E C Z by
(b).
and
Thus
B @ s
full
sub
such
that
A C C.
C be a f u n c t o r
Z C IA,V] m.
E
s @ A E Z.
in Z~ as r e q u i r e d .
(c) ~
B(B,F)
isomorphisms
for all
(c).
~ @ B are
category
If C E C Z t h e n
isomorphisms
C.
~ C(s,C/B)
Let
exponentiation
in C Z for all B ~ C and
(b) ~ C(
[email protected],C)
(b).
are
s\C are
generates
C/B b o t h
and
(a) ~
~ C(A,S\C)
strongly
under
A ® s E Z and
is m o n o i d a l .
Proof. C(
[email protected],C)
s E Z imply
category
form
(as is done
consisting
for all B E B.
model
of the
of A in B.
of the
[A,V]
form
in [13],
§8.1)
functors
F E
Such a functor Clearly
and
let the
[A,B]
F is i n t e r p r e t e d
as a
[A,B] Z = [ A , B  U = L A , B ] z
for
Z c U c ~. We now
egory
over
which
the
category
suppose
V and
that
internalhem
convolution
functor
B is a m o n o i d a l functors
is a small
biclosed
defined
by
promonoidal
category
(5.2)
exist
over
cat
V for
on the
functor
6.2.
If Z is a m o n o i d a l
category
A,V]
then
class
of m o r p h i s m s
IA,B] Z is c l o s e d
under
in the exponen
in [A,B] . Proof.
have:
A = (A,P,J,...)
[A,B] . Proposition
tiation
that
For
all
s C Z, B E B, F E [A,B]
and
G E [ A , B ] z , we
42 J" [ s x , B ( B , ( G / F ) ( X ) ) ] X / [Sx,B(B , f H ( Y , F Y , G ) ( X ) ) ] X Y
by (5.2),
~ ~ / [ S x , B ( B , H ( Y , F Y , G ) ( X ) ) ] Y X by interchanging
limits,
/ / [Sx, YX /
(/ Y
by interchanging
/ [P(XYA),B(B,GA/FY)]] A
by ( 5 . 1 ) ,
[ / X s X ® P(XYA),B(B®FY,GA)]) A
limits and using the tensorhom
adjunctions
of V and
B, f (f [(s ® A(Y,))(A),B(B®FY,GA)]) Y A
(*)
because
(s @ A(Y,))(A)
= /
XX'
s X @ A(Y,X') @ P(XX~) by ( 1 . 2 ) ,
X
/ by t h e
representation
theorem
end o v e r Y E A o f i s o m o r p h i s m s dal
and G E [A,B]z
i n [A,B] Z f o r
all
Corollary
applied because
by hypothesis. F ~ [A,B] 6.3.
s X ® P(XYA)
The m o r p h i s m ( * )
s @ A(Y,)
E Z since dually,
is monoidal
If Z is a class of morphisms
6.1.
in the convolution
Thus [A,B]z,
is closed under exponentiation Thus,
if the Zcontinuous
A form an exponentially
biclosed
then so do the Zcontinuous oidal biclosed
in
category
an
Z is monoi
F\G b o t h l i e
in [A,B]
models
for which
functor cate
in [A,V]
then
which coincides with by Proposition
in V of a promonoidal
subcategory
models
[A,V]
in [A,B] .
If [A,V] Z is closed under exponentiation by Proposition
is
and G E [A,B]z.
then [A,B] Z is closed under exponentiation
Proof.
[A,B]~,
~ E A.
Thus G/F a n d ,
[A,V] Z is closed under exponentiation gory [A,V]
to
of the convolution
6.2. category [A,V]
of A in any suitably complete mon
B which is based on V.
43
This result in [2], Chapter III.
contains an a l t e r n a t i w For example
be the simplicial category. gories and functors
app:~oacn to c o n s t r u c t i o n s
(c~~. [2] ~12), let V = S and let
Then the category Cat of small ca~e.
is fully embedded in [~°6~S]
r e f l e c t i v e subcategory.
as a c a r t e s i a n cl6sed
This implies that C~t is defined by a (carte
sian) m o n o i d a l class of m o r p h i s m s
in [A°P,s]]
m o r p h i s m s inverted by the reflection.
namely,
~he class of all
It ne~ ~ follows from C o r o l l a r y
6.3 that the category [~°P,B] Z of all category objects in B is closed under e x p o n e n t i a t i o n in [~°P,B]
w h e n e v e r [~°P,B]
hom functors
if the m o n o i d a l b i c l o s e d
(5.2).
Moreover,
c o n s i d e r e d in §5 exists on [~°P,B]
admits the i n t e r n a l structure
and the e m b e d d i n g
[~°P,BIz C [~°P,B 1 has a left adjoint then this a d j u n c t i o n admits m o n o i d a l e n r i c ~ e n t the r e f l e c t i o n theorem of §i.
by
Note that, by P r o p o s i t i o n 6.1, Z may be
r e p l a c e d by an~ class of m o r p h i s m s in [A°P,s]
w h i c h defines Cat as its
class of orthogonal objects. Remark.
The general q u e s t i o n of the e x i s t e n c e of a leftadj
oint functor to an i n c l u s i o n of the form A,B] Z C [A,B] studied in some de
L2.
If
L3.
If @~ C(x)
Example
~
This class is stable
TONz
The category
spaces and continuous maps.
[3]) that a limit space is a set
xcX~
C(x)
bEB
a set
C(x) of "convergent"
satisfy the axioms
X
filters
:
C(x).
@>.~ and ¢c C(x)
3.2.
and ~
Let Z
then @E C(x)
C(x)
be the
then 0aCe C(x)
(larger)
generated by the i d e n t i f i c a t i o n maps
class of m o r p h i s m s
f:A
÷
B
with the following
property: for each point
bEB
and open covering
exists a finite set is a n e i g h b o u r h o o d category
TONz
{Xl,...,~n}CA of
beB
<x>
c
of
flb
there
fGxlU .ufGxn The r e s u l t i n g
and ¢ c C(x)
then 0 E C(x)
if there exists a set of filters
{ @X ; ~ A {V X ; XcA {V~.I ;
:
C(x)
@ E C(x)
The class
XeA}
is equivalent to the category of those limit spaces
L2. If @~¢ L3'
such that
(see [5] T h e o r e m i).
which satisfy the following axioms LI.
{G X ;
and @~c and
} such that each set
V X E @~} contains a finite subset
i = l,'',,n} Z of m o r p h i s m s
interior of the r e f l e c t i o n from smaller than the m o n o i d a l
C(x)
such that
UV~i ~ @.
obtained here is the stable To N
interior
to
Top
(see [5]
§3
and it is strictly for a counterexample).
72
Finally, we note that in order to completely describe the canonical Grothendleck topology on classes
{fa : Am ÷
given space property: ~
B
Top
it is necessary to consider all covering
B ; ~ ~}
in
Top
Such a class of morphisms into a
.
covers in the canonical topology if and only if it has the
for each point
b~B
and open covering
there exists a finite subset
is a neighbourhood of
{Gl ~ ~ A
{ll''"'In } c A
and
such that
AeS}
of
edife(Gll)
b~B . However, it is easy to verify that the introduction
of these covers imposes no new restriction on the limit spaces under consideration. In other words it suffices to take ~
small so that
~
is generated by the
identiflcationmaps already described above.
4.
Stability conditions and relations. In this section we note several "known" facts about general closed span
categories.
Proposition 4.1. and only if
Proof.
SpanC
C
is a closed span category if
is closed as a bicategory.
For each morphlsm
pullbackalongf SpanC
A finitely complete categor~j
and let
fcC(BC)
f, : C/B
is closed if and only if
f*
let
~ C/C
f* : C/C
C/B denote
denote the left adjoint of
has a right adjoint
result then follows frown the fact that if
+
[f,]~
a f
~f
f~ ~C/B
'
C / c
f . The
exists then it has a
unique llmltpreserving extension along the comonadlc functor
C/C ~
for all
f* .
f, :
73
Suppose
B
is a
cscategory containing a strongly generating class
such that each
B~B
has a presentation as a coequaliser:
2A
B .
A class
x
all
~
EA
s~Z
~
y and
f : A
Proposition 4.2.
÷
B
Z
of morphisms in
with
=
A ~ A . Let
z(A) if
z
is Astable if f*s~Z
~
(b)
If Z = Z
(c)
If
Z
consists of epimorphisms then
if
Z
is stable.
Z
for
AeA}
Z(A) = {s: B ÷ A: s~Z and
(a)
then
is
B
A
Astable.
is stable if
Z
is
Astable. Z
is stable
The verifications are straightforward. If then
Z
Z
is a stable class of morphlsms in a finitely complete category
is closed with respect to span composition.
{B/AxB ; A,BEB}
categories of fractions of
Thus the class of
with respect to
forms a new
bicategory
(cf. [7] )
suppose
is a closed span category with a proper factorisation system
where
B E
is stable.
Mrelations in
B .
with the evident universal property.
Z
Then, on takirg Z = E, Because
SpanB
is closed under exponentiation in
In particular,
EM
one obtains the bicategory of
is closed the subbicategory of relations SpanB
(by the "several objects" form or
the reflection theorem for closed categories [6] )
and the exponentiation
provides a form of universal quantification (cf. [ll] ).
Bicategories (not necessarily closed) of relations and spans have also been considered in
[12]
by J. Meisen,
74 References
[ I]
Antoine, P., Extension minimale de la
[email protected] des espaces topologiques, C.R. Acad. Sc. Paris, t.262 (1966), 13891392.
[ 2]
[email protected], J., Introduction to bicategories, Reports of the Midwest Category Seminar I, Lecture Notes 47 (Springer 1967), 177.
[ 3]
Binz, E. and Keller, H.H., Funktionenr~u~neinder Kategorie der Limesrgume, Annales Acad. Sc. Fen., A.I.383 (1966), 421.
[ 4]
Day, B.J., Relationship of Spanier's quasltopological spaces to kspaces, M.Sc. Thesis, Univ. Sydney, 1968.
[ 5]
Day, B.J. and Kelly, G.M., On topological quotient maps preserved by pullbacks or products, Proc. Cambridge Phil. Soc. 67 (1970), 553558.
[ 6]
Day, B.J., A reflection theorem for closed categories, J. Pure and Appl. Alg., Vol.2, No.l (1972), iIi.
[ 7]
Day, B.J., Note on monoidal localisation, Bull. Austral. Math. Soc., Vol. 8 (1973), 116.
[ 8]
Day, B.J., On adjointfunctor factorisation, Proc. Sydney Cat. Conf., to appear.
[ 9]
Fakir, S., Monade idempotente
[email protected] ~ une monade, C.R. Acad. Sc. Paris, t.270 (1970), 99101.
[I0] Freyd, P. and Kelly, G.M., Categories of continuous functors I, J. Pure and Appl. Alg., Vol.2, No.3 (1972), 169191. [ii] Lawvere, F.W., ed., Toposes, algebraic geometry and logic, Lecture Notes 274 (Springer 1972), Introduction and references. [12] Meisen, J., On bicategories of relations and pullback spans, preprint, University of B.C., (1973).
REVIEW
OF THE E L E M E N T S .............
OF 2  C A T E G O R I E S
H.'.......
by G.M. The purpose duction
ion,
notions
to avoid
original,
ary facts below
papers
needed
by each of us,
chiefly
to i n t r o d u c e
pasting
that we use
our n o t a t i o n
to give a treatment,
which
seen,
adjunctions
fJ u and f'Ju'
ality.
and then m e n t i o n the
2category
i.I
DOUBLE
Both notions first
Cat, Set,
respectively
egory
the notion
categories,
is a c a t e g o r y
elementary
of these
CATEGORIES
of d0uble
the most
of m a k i n g
in CAT;
element
our papers
the o p e r a t i o n
of
operation
complete
than
arising
from
= (f'b,af)
and of its natur
of monads
that
but
become
it is really
in a 2category, available
in
a 3category).
AND 2  C A T E G O R I E S due to Ehresmann; category.
the categories
object
is
in the literature
and more
2category,
(because
and of small
see [6j and [ 71.
We denote
of all categories,
sets. but
Conceptually, it admits
the
by CAT, of all
a double
SET,
sets, cat
following
description.
It has objects arrows
(bu,u'a)
propemties
are o r i g i n a l l y
We recall
of small
in any
some e n r i c h m e n t s
§i
here
care to read them,
to us simpler
the basic
of 2  c a t e g o r i e s
as may
and e s p e c i a l l y
of the i s o m o r p h i s m
In §3 we recall
account
together
some notat
In §2 we use the p a s t i n g
seems
any we have
of substance
in the hope
for such beginners
intro
by c o l l e c t i n g
In §I we rehearse
partly
constantly.
as a common
and e s t a b l i s h i n g
Nothing
our needs.
2categories,
selfcontained
is to serve
find no c o n n e c t e d
satisfied
about
Str~t
in this volume,
later duplication.
but we could
that e x a c t l y
{~d Ross
of this r e v i e w
to the authors'
some basic
Kellv
A etc.; h o r i z o n t a l
x etc.; and squares
ain functions
sufficiently
~ etc.;
there
indicated
arrows
a etc.; v e r t i c a l
are various
by the diagrams
domain
and codom
76
a A
~B
A
A
~B
a x
Y
C
C
b
mD
The objects and the horizontal arrows form a category, with identities hA: A ~ A ;
the objects and the vertical arrows form a category, with A $ VA
identities
The squares have horizontal and vertical laws of
A composition, represented by A
x:
a ~B
c 7
c dE
D
~F
lz
x
A
a
C
,
~ B
~D
u G
e
~H
;
under each of these laws they form a category, with respective identities A x[
hA
~A
ix
C
A Ix C
,
a
VA I
la
A
a
:B IVB
In the situation
the result of composing first horizontally and then vertically is to be the same as the result of composing first vertically and then horizontally. The composite
77
A
a ~B
A
is to be ica, identities.
and
Finally
~B
a
similarly
c ~E
r E
c
for v e r t i c a l
the h o r i z o n t a l
composition
and v e r t i c a l
hA ~A
1
identities
hA
A
A
of h o r i z o n t a l
A
~
~ A
i
,
A ~ A
hA
,
hA
are to coincide. Examples and in §2.2
1.2.
of double
categories
may be found
in P a l m q u i s t
21],
below.
A 2category
K may be thought
w h i c h all the v e r t i c a l
arrows
are
of as a double
identities.
category
A more
direct
in
descrip
tion is as follows. K has qbjects 1cells
f: A ~ B etc.,
or 0cells and
A etc.,
arrows
or m o r p h i s m s
or
2cells f
A
~ v
_ e ~
B
or
A
$ ~
g (Kelly
tends
to
it is purely
use
g
double
a matter
The objects underlying
category
shows what
is meant,
For them
arrows
for
2cells,
and the arrows of K, with
Street
form a category
identities
we sometimes
write
fixed A and B, the arrows
composition:
under the
/ A
composite
and
single
ones

of taste.)
form a category K(A,B)
The v e r ~ c a l
B
f ~
~
~
~ ~
h above
_
K0, called
IA: A ~ A; when the
the context
K for K 0. A ~ B and the
operation
known
2cells
between
as v e r t i c a l
~ B.
is d e n o t e d
by ~.a or B.~,
or rarely
78 by 8~ when no c o n f u s i o n introduced
below;
is likely
its identities
with the h o r i z o n t a l are denoted
composite
to be
by
f
f There whereby
is
also
a law of
horizontal
composition
U
B
B
g
V
a 2cell f
u
uf =
g this
2cells,
from 2cells f
we g e t
of
v
composite
2cells
C;
y*~
vg
is also denoted
by ¥~:
are to form a category, with iA A ~
We r e q u i r e
finally
A
uf ~ vg.
Under
this
law the
identities
.
1A that,
in
the
~ •
situation
...... ~ C ,
h the
composites
(6,8).(~*a)
and
(6.y)*(~.~)
coincide;
and that
in the
situation
we have
lu*l f
=
We also object arrow
f
U
f
u
luf. freely
use the c o n v e n t i o n
whereby
A or of an a r r o w f is also used as the name i A or its identity
composite
2cell
if.
In p a r t i c u l a r
the name
of an
of its identity the h o r i z o n t a l
79
A
B
C
f is a l s o
written
D
v
g
as U
A
~
B
~
C
=D
~
f
g V
and denoted
by gyf.
The more
general
basic
above
operations
operation
situations
on 2  c e l l s
of p a s t i n g ,
can be c o m b i n e d
introduced
by B e n a b o u
to give [i] .
the
The
two
are
V
The
first
and
the
give
of t h e s e
second
meaning
identity
has
2cell One
to g i v e
to i n d i c a t e
to s u c h
If in a d i a g r a m
one r e g i o n
is m e a n t
the
2cell
composites
the
vy.6f:
2cell
Bg.u~:
uf ~ uhg ~vg,
uf ~ v k f ~ vg.
Thus
we
as
s u c h as
no
2cell
is m e a n t ,
to
f
g
h ~
k
marked which
can g e n e r a l i z e
meaning
to i n d i c a t e
in it, implies
the p a s t i n g
such multiple
it is to be u n d e r s t o o d that
the
vf = hu.
operation
composites
that
as
further
still,
so as
B0
This
is meant
of the
to indicate
a vertical
composite
of h o r i z o n t a l
composites
form
r
there
is usually
a choice
taken,
but the result
simple
cases,
1.3.
is independent
and can be proved
after an a p p r o p r i a t e itions
of the order
of this
in terms
the
choice;
i n d u c t i ve l y
formalization
composites this
is clear
in the general of p o l y g o n a l
are in
case
decompos
of the disk.
As for examples
of 2categories,
just as the p a r a d i g m a t i c
category
the
and the
arrows
are
The context merely
in which
functors,
will
the p a r a d i g m a t i c
is SET. 2cells
The objects are natural
show when CAT is c o n s i d e r e d
the u n d e r l y i n g
category
is meant.
one
are categories,
transformations.
as a 2category
There
is CAT,
is the
and when
sub2category
Cat of small categories. For a m o n o i d a l Vcategories, of E81;
Vfunctors,
a g a i n with
categories
the
themselves,
transformations The a 2category
category and
Vnatural
sub2category with m o n o i d a l
in the sense
category
V, we have
of L8J,
K of ordered
when we observe
that
the
2category
transformations,
in the
VCat of small ones. functors
and m o n o i d a l
form a 2  c a t e g o r y
objects
K(A,B)
sense
Monoidal natural
Mon CAT.
in any c a t e g o r y
the homset
VCAT of
A becomes
has a natural
81
order,
and can therefore For a c a t e g o r y
be r e g a r d e d A, the comma
which
is a category
B together
which
from
(C,C)
(B,B)
a 2category natural
transformations will
1.4.
Besides
by 2  f u n c t o r
oint;
see [141
do not given
use
and by
2cells
B ~ A, and an arrow of
B ~ C with
CT = B, becomes
C~ : id.
Many
of 2  c a t e g o r y one:
K is Just
other
treatment
of
§1.2 above, CAT is
CATcategory, determines
transformation:
Similarly
in
the category
a
This d e f i n i t i o n
or "2adjoint"
assigns
terms,
of L, arrows
what
namely
Vadjunction.
its we shall
CAYfunctor
for 2adjoint,
in the more
domains
and composition. to each object
is not only natural nB.Df ~:
a 2functor
of K to arrows
of L, p r e s e r v i n g
identity
(1.1)
of
: CAYadjNote
general
that we
senses
to t h e m by Gray in [lO].
to objects
2cell
definition
transformation.
"2natural"
B:
an object
below.
2natural
for a general
CAT/A,
a: T ~ S: B ~ C to be the
but n o n  e l e m e n t a r y
K(A,B).
In e l e m e n t a r y
have
T:
~: T ~ S for w h i c h
and a 2  c a t e g o r y
hom b e i n g
and C A T  n a t u r a l
2cells
the e l e m e n t a r y
closed,
CATvalued
is a functor
in the papers
is an e q u i v a l e n t
cartesian
mean
arise
category
with a functor
w h e n we take the
examples
there
to
as a category.
A 2natural
A of K an a r r o w
but also
2natural
EB
category;
2cells
of K to
and all types
transformation ~A or HA: that,
in the
of
n: D ~ E:
K ~ L
DA ~ EA in L, which for
f: A ~ B, we
sense
that,
for each
= DA
Eg
As in general D to the
V, so here where
of L, and
of K
Ef
Dg
Vfunctor
objects
sense
f ~ g in K, we have Df
DA
L sends
and codomains
in the o r d i n a r y
= El.hA,
D: K ~
the
Vnatural
Vfunctor
transformations
E form not
V = CAT the 2natural
in other words
from the
only a set but an object transformations
2CAT is really a 3category,
of
D ~ E form a i.e.
a 2CAT
82
category.
We
follow
transformations
Benabou
[i]
modifications.
in calling
morphisms
of 2natural
A modification
p: q * ~ :
K "* L ,
D ~ E:
also written
D
E assigns
to
each object
A of K a 2cell
pA: nA ~ ~A s u c h t h a t ,
for
f: A ~ B, we h a v e nA
nB
(l.2)
DA ~
DB
~pB
EB = DA
~pA
~B Zn p a r t i c u l a r , gory End
K of
its
2cells
this
2category
The what
reverse dual,
Observe
Kelly feels
1cells
2category
of K; its
objects
like
by the
that
K c°°p
i s m of c a t e g o r i e s Specializing
transformations
P: n ~ ~.
When working
the r i g h t
definition we h a v e
not
the
to w r i t e
q:
q:
D ~ E; a n d
totally
within
D ~ E and
of a 2category
K°P(A,B)
2cells.
duality
thus we reverse
are
2functors
D:
CATadjoint;
K(EB,A)
the
K also deter
= K(B,A),
We w r i t e
on CAT,
to the
is j u s t
K ~ that
~ L(B,DA)
of [14] same
on
thing
i ~ DE a n d ¢: E D ~ i s a t i s f y i n g
2adjunction
endo2functors
it.
ordinary
the r e s u l t
it c o m e s
are
a 2cate
so t h a t
we
K c° f o r the o t h e r
so that
2cells
but
not
the
1cells
= K °pc°.
say t h a t they
K, we g e t
2natural
reserves
but
= K ( A , B ) °p"
to say t h a t
q:
a fixed
are
K °p s h a l l m e a n :
the
To
see t h a t
arrows
nonelementary
induced
KC°(A,B)
taking
are m o d i f i c a t i o n s
p: q ~ ~ if he
mines
~A
endomorphisms
D: K ~ K of K; its
EA      ~  ~ S S .
/
adjunction
L a n d E: is,
which
that
there
is 2  n a t u r a l
Vadjunction to h a v e the u s u a l
i__n_nthe
L ~ K are
to the
2natural
is
is a n i s o m o r p h in A a n d B. c a s e V = CAT,
we
transformations
conditions;
2category
2adjoint
2CAT,
so t h a t in t h e
sense
83
of §2.1 below. i. 5 .
A great many
notions
2CAT g e n e r a l i z a t i o n s
within
in the basic a given
isomorphism phisms
notions"
do indeed
indicate
before
this
a systematic
theorem"
shows
replaced
one)
in the
nomenclature
the
[22],
and are
[i].
Indeed
egory,
these
We should
notion
morphism:
choices
has
criterion
2cells
case
to
in others by the
illustrates notion,
a general
the d e f i n i t i o n
as the norm, replaced
by an
principle
we call the
N".
and
D: K ~ L b e t w e e n IDA ~ DI A instead [24]
morphisms
of b i c a t e ~ o r i e s
point:
to
but
by Street
sense
we
n o t i o n with
of this morphism,
of lax functor
of B e n a b o u ' s
like to
use.
relaxed
Dg.Df ~ D(gf)
here;
(doubtless
however
N is taken
an "oplax
in B e n a b o u ' s
book
of usefulness,
have b e e n c o n s i d e r e d
a bicategory
"morphismlike"
seem in fact
we have no general
of the sense
sense r e v e r s e d
a special
and this
"relaxed
consider;
notion with e q u a l i t y
is the concept
which
of equalities;
they
isomor
some
systematically
that we shall
"strict"
by a mere
An example 2categories,
these
forthcoming
of Gray [12].
c h o s e n a sense by some
same n o t i o n with
this
them to be r e p l a c e d
to go into this
in detail
for the r e l a x e d
offer for our various having
only to
the given
that
willynilly
a l l o w in g
was to
to satisfy
in some cases
isomorphism, and "lax N" for the still more equal i t y
a diagram
sense);
be r e q u i r e d
that we must
the original
say "pseudoN"
(in a given
experience
our i n t e n t i o n
are treated
Where
For where
notion.
It is not such things
V;
and replace
in nature;
a "coherence
for an a r b i t r a r y
or even go further
Moreover
thing,
in
it to commute
then usually
occur
normal
"strict"
to appear
will
axioms".
we can prove
no sense
for any V admit
we can now allow
by a given m o r p h i s m
be the more
simpler
notion,
isomorphism,
or m o r p h i s m s
"coherence
that w o u l d make
2CAT is a 3category.
and this b e c a u s e commute
in VCAT
definable
is itself when
continues
and Roberts
a pseudo2cat
one relaxes to make
sense
some when
84
the domain
and the c o d o m a i n
such do not occur the p a p e r ~25]
paper
of Street
[i0]
of 2natural
of Kelly and
below,
[15],
pseudomonads
there
are the lax algebras
their
lax morphisms...
with
strict
and
and i d e n t i t y
where
take the lax n o t i o n "strong
strict
functor"
(with an i s o m o r p h i s m
functor"
(with e q u a l i t y
2.1.
Other accounts
pasting (2.2)
arrows
by Gray
[ 2].
[3] ; the 127].
the
Then and
"strict
Even
lax m o r p h i s m s
special
the tensor
N; we then
in
Another
over these,
for them,
case
of
product
In these
cases we
call the pseudo
N";
thus
"monoidal
"strong m o n o i d a l
CA @ CB ~ @(
[email protected])),
here;
such lax or pseudo papers
ADJUNCTION
and
"strict
monoidal
in [19],
as we need are
IN A 2CATEGORY
below, [14],
of v a r y i n g
and [211.
of the a d j u n c t i o n
from R.F.C. ~,s:
notions
below.
of the m a t t e r
for the neat e x p r e s s i o n
An & d j u n c t i o n
in the
CA @ @B = ¢(A®B)).
can be found
b e l o w we learnt
below
Bunge
in nature.
notion
El31.
of our nomenclature.
( p re s e r v i n g
notion
in
transformation
of Bunge
CA @ @B ~ ¢(A®B)),
in the i n d i v i d u a l
completeness,
ones
N" and the strict
§2.
in
as
occur
"2natural"
as is shown by the
are quite rare
(with a c o m p a r i s o n
functors
by Z o b e r l e i n
aspect
algebras
thing,
functor"
introduced
by Gray)
2monad)
second
as our basic
We say no more
called
are c o n s i d e r e d
strict
on the nose)
do occur
or the p s e u d o  a l g e b r a s
are the normal
monoidal functors,
notion
(relaxed
us to the
2monads,
of al g e b r a s
of lax natural these
(now adopted
corresponding
Lax
but p s e u d o  f u n c t o r s
and are the things
of lax m o n a d
"lax".
and go back to G r o t h e n d i e c k
transformation;
is the notion
brings
book,
is the n o t i o n
"quaslnatural"
This
themselves
in the present
A second example in place
are
degrees
of
The utility
equations
of
(2.1)
and
Walters.
f4u: A ~ B in a 2category
u: A ~ B and f: B ~ A t o g e t h e r
with
2cells
K consists
~: i ~ uf and
of
85
c: fu ~ i s a t i s f y i n g
i
A
(2.1)
the axioms
~A
equals
identity,
B
~, 1
I
equals identity
B We
i
say that
" ~"B
f is left adjoint
we call ~ the u n i t , a n d When
Vnatural usual
s the counit,
K is VCAT
shown by Kelly
in 14]
of a d j u n c t i o n
If ~1,si: a composite
for a symmetric
A(fb,a)
monoidal
fi~ui:
adjunction
in p a r t i c u l a r
V, it has been with
we get the
K = CAT.
B ~ C is a second
2,c2:
closed
f4u are in b i j e c t i o n
~ B(b,ua);
when
to f;
of the adjunction.
that a d j u n c t i o n s
isomorphisms
notion
to u, and that u is r isht adjoint
ffl~ ulu:
adjunction,
we clearly
A ~ C if we define
g~
q2,e2
as the composites i
(2.3)
A
/B~r~l
1'
A
,L B
.....
~A
S
Ul\~ c / fl ~
C
~
.
i
Thus
adjunctions
in K form a category,
with
i,i:
i~i:
A ~ A as
identities. 2.2. and E
If we look upon are m u t u a l l y
the f o l l o w i n g
(2.1)
inverse
proposition
and
(2.2)
under
becomes
the
as a s s e r t i n g indicated
evident:
that
pasting
the
2cells
operations,
86
Proposition n',s':
f'~u':
a bisection
(2.4)
2.1.
Let
A' ~ B'.
between
is t h e
D,s:
Let
2cells
a: A ~ A'
and
b:
B ~ B'.
2cells
a
compo,s,i,te
B
X is the
A ~ B and
I: bu ~ u ' a a n d
A
(2.5)
f~u:
~
A T
~B
...........
b
~B
i
composite
~:
f'b ~ af,
expressed objects
naturality
~A
as
are
follows. those
the v e r t i c a l tal
arrows
arrows
,
~A'
double ~,s:
is j u s t
category,
f~u:
the
both
double
are
same
given
2cells
in
sides
Proposition isomorphism That
between
is to say~
vertical
(2.4).
In the
horizontally
2.2.
The
above
and horizontal.
of horizon
above.
(In o u r In the
first
b: B ~ B'
and vertical
is a 2  c e l l category,
X in K
a square
in
(2.5).
composition
or v e r t i c a l l y , is n o w
the
of squares
expressed
between
I and
categories
we h a v e
just
composition
and
In
corresponding
bijection
respects
o f K, a n d
of v e r t i c a l
"oblique".)
double
in q u e s t i o n
the two d o u b l e
the bijection
as
be
In e a c h the
the a r r o w s
~ in K s u c h as a p p e a r s
horizontal
"naturality"
§2.1
A' ~ B '
second
~ may
composition
of
B'
Composition
a: A ~ A'
f'~u':
X and
are
in K.
shown
sides
is a 2  c e l l
by p a s t i n g , The
with
and ~',~':
categories,
of K.
arrows
of a d j u n c t i o n s
=
categories.
in K, w h i l e
is c o n v e n i e n t l y
a square
A ~ B,
s u c h as a p p e a r s with
composition
T
i
between
two d o u b l e
horizontal
B T
~
b
are the adjunctions
composition
"vertical"
'
bijection
Consider
o f K, the
arrows
is the
diagrams
of this
where
AT
~
T
The
is
a
A
B
there
i
..........
i
Then
by: ~ is an
described.
identities,
87
Proof.
As regards
are r e s p e c t i v e l y by
(2.4)
only
that
i,I:
= I by
I~i:
identities:
A ~ A and i,i:
~ = I and in p a r t i c u l a r
if k = i.
identities,
vertical
For h o r i z o n t a l
identities,
and f~u c o i n c i d i n g
(2.4)
and
(2.2),
For vertical
that,
with
I~i:
then
toget h e r
two diagrams
and observe
that
It will mates
under
mention
and
we have
only
to write
f (2.4)
the central usually,
we paste
to get
triangles
s' and ~' cancel
be u n a m b i g u o u s
f~u and
f'~u',
out by
if we call
without
(2.1).D k and
explicit
2.3.
l_f f~u an d f,~u
then
f and f' are
isomorphic.
Proof.
Let
the mates
of I
A
B
f'~u
composition
and b.
Proposition
under
D
For h o r i z o n t a l
in context,
the adj.unqtions
of a
canonically
(2.4).
like
if I = I we have
if ~ = i then k = I.
....
(2.3)
A' ~ A', we have
we have a and b both
/.\. and to look at
and f'~ u'
when a = b, ~ = i if and
f'~u';
and s i m i l a r l y
composition
if f  ~ u
the a d j u n e t i o n s and f~u,
f~u
~A
1
"~B
and f'~u,
be r e s p e c t i v e l y
and under
the a d j u n c t i o n s
88 i A
D A
B ..........
T h e n by t h e
~B
i
"horizontal"
i A .............
and
part
B
of Proposition
~
I
2.2,
m A
B
p and v are mutually
inverse. 2.3.
If D:
K ~
in K c l e a r l y applying
L is a 2  f u n c t o r ,
gives
D to
an a d j u n c t i o n
(2.4)
Propositio..n
f~ u a n d
2..4.
f'J u'
Dp a r e m a t e s
when
(2.5)
D~,Da:
p are
and
if D:
Df~ Du:
2.5.
~: D ~ E b e a 2  n a t u r a l DA
mates
Df~Du
in g e n e r a l
~A
Let
DA ~ DB in
und.e.r., t h e
and
L.
By
ad~unctions
Df'~Du'
true
that
D,E:
K ~
[ be
2functors
transformation. , EA
"
DB ................
Then
in
~ is the
under
Proof.
~
EB
L. D
identity
,
and
let
identity ~A
DB
~
2cells
~ EA
EB
,
~B
the . adSunctions The
the
DA
~B
above
A ~ B
k is; but we h a v e :
Proposition
are mates
f~u:
K * [ is a 2fu,n,c,,t,o,,r~ , t h e n DI a n d
the a d J u n c t i o n s
it is not
~,¢:
we get:
I_~f t
in K, a n d
under
Of course 2cell
and
an a d j u n c t i o n
2naturality
Df~Du
and Ef~Eu
of ~ expressed
in
in the
L. form
(I.i)
gives ~A D A ...............
DB 
~
~ EA
DB .............
equals
~ ~B
EB
~A
DA
~EA
DB ~B
~' EB
1
~
EB;
89
pasting the
E~
form
on the r i g h t
of each and
The notion
MONADS
of m o n a d
object
B of K, is a n e n d o m o r p h i s m
i ~ t,
o f the m o n a d ~tn
A detailed Street
t; = i,
in
~.t~
generality
case
where
o f the
2category
in K, on the
with
and
2cells
the m u l t i p l i c equations
= ~~t. has b e e n
simpler
given
aspects
and
by
then
K = 2CAT.
of the m o n a d
t above
on an a r r o w
s: A ~ B is a
v is c a l l e d
a talgebra
v: ts ~ s s a t i s f y i n g = i,
s together
A).
s ~ s'
vtv
with
A morphism
such
= v~s. an action
o__fft  a l g e b r a s
(with
common
domain
A)
(with
is a
such that
(3.3)
v'.t~ the
= ov.
talgebras
forgetful
functor
~r:
t 2 r ~ tr~
tp:
t r ~ tr'
FA:
K(A,B)
talgebra
with
talgebras
A form a category
sending
A ~ B it is c l e a r
is a t  a l g e b r a ; is a m o r p h i s m
and that
(s,v) from
It is f u r t h e r
s, the a c t i o n
v: ts ~ s is a l s o
following tr free
(3.1)
of t  a l g e b r a s .
the a c t i o n proposition,
talgebras:
~s.
in the
that
from
these
light
tr,
gives
with action
a functor
(3.2)
a morphism
From
with a
O: r ~ r',
This
clear
A/g(A,t),
to s a n d ~ to i t s e l f .
for any
~ A/g(A,t).
ts is t a k e n w i t h the
domain
U A to K ( A , B )
F o r a n y r:
verify
we w a n t
in any
the u s u a l
= i,
some
An arrow
when
what
A monad
the u n i t
satisfy
we r e c a l l
v.~s
Thus
~~t
to
in 123] ; h e r e
(3.2)
a:
respectively are
sense
t: B ~ B t o g e t h e r
in this
special
makes
K ~ CAT.
of m o n a d s
An a c t i o n
domain
these
where
treatment
to the
2cell
is that
~: t 2 ~ t c a l l e d
(3.1)
pass
case
(= t r i p l e )
classical
ation
gives
IN A 2  C A T E G O R Y
K; the
n:
(2.2)
(2.4).
§3 3.1
using
that,
for a
of t  a l g e b r a s ,
remarks
of which
we e a s i l y we
call
the
9O
Proposition
3.1
s: A "~ B be a t  a l ~ e b r a bijection $iven
between
to U A.
a n d r: A ~ B any arrpw.
2cells
c
In the commonly
of B, a n d A/g(~,t) by B t.
:
In d e t a i l ~
Then
~: r ~ s a n d t  a l ~ e b r a
vt~,
s: ~ ~ B is then v: ts ~ s,
Identifying in this
course
in the
general
K(A,t)
on the
category
K~A,B) K(A't)
there
let
is a
morphisms
~:
tr ~ s,
with
A, t h e n
with
primary
with
case
case
the
sense
B, we w r i t e
as ut:
the m o n a d
K(A,B),
is a n o t h e r
t ~ t'
such
denotes
ZT:
clearly
(s,v)
(3.6)
and
category
The
adjoint ft:
object
functors
the
U~
Of
a classical
is just
s
denoted
B ~ B t.
t on B i n d u c e s
The
category
is c l a s s i c a l l y
B t ~ B and
Alg(A,t)
~.
corresponding in B.
the
is m o r e
monad
category
monad
on the
same
B, a m a p
of m o n a d s
This
~:
tt ~ t't'.
q' = T.q; If
is a t  a l g e b r a
(S,V')
is a t '  a l g e b r a
with
where
s ~ s' of t '  a l g e b r a s
gives
is a
that
v = v'.~s;
a morphism
talgebras.
the unit
s ~ s' are m o r p h i s m s
~''T 2 = TH, 2
domain
"talgebra"
of K ( A , t )  a l g e b r a s .
(3.5) T
~.nr.
K : CAT the p h r a s e
in this
classical
If t' T:
~:
=
identified
K(~,B)
and F !
2cell
case
to t h o s e
of t  a l g e b r a s
3.2
a
classical
restricted
talgebra
and
a dJoint
by
(3.4)
here
F A is left
a functor
Alg(A,t ' )
A/g(A,T)
Alg(A,T)
(3.7)
K(A,B).
is also
a morphism
rendering
of
commutative
~ A/g(A,t)
domain
91
In particular the
the
t'
itself
is a t  a l g e b r a
under
action
(3.8)
8: tt'
a n d T c a n be r e c o v e r e d (3.9)
T: t
since
~''t'n'
given,
when
= i.
t't'
(3.9)
and
with
8.t~'
immediate
that
= f'~'u'
f'q u'. (3.11)
Proposition 2cells
v:
3.3 every
is
~ t' be a t  a l g e b r a m o r p h i s m ,
that
is, t h a t
u'
on B w h e r e senerated
is a t '  a l g e b r a
t'u'
= u'f'u' and
3.2.
with
t'
= u'f'
by t h e
and
adjunction
action
~ u'
if t is a n y m o n a d
between
in K, it is
2cells
o n B, t h e r e
T: t ~ t'
= u'f'
is by
and
by T = vf'.t~'
T is a m a p
of m o n a d s
if a n d o n l y
if v i s
of t o n u'. P~'pof.
(3.11)
and
If T is a m a p thus
o f t on u',
trivially
of t on t'
for it to arise as in (3.8)
C ~ B is any a d j u n c t i o n
v = u'~'Tu',
an a c t i o n
t't'
St';
the m o n a d
given
Proposition
~':
taction
is t h i s m o n a d
(3.12)
action
this
that
~ u',
0: tt' ~ t'
condition
is a m o n a d
2.1 a b i j e c t i o n tu'
action
= ~'.St'
u'~': I f t'
is t h a t
f'q u':
We c a l l
form
~ t', 8
the
(t',~',~')
Observe
8 in the
sufficient
of m o n a d s
If ~ ' , ~ ' :
u' b y
from
~ t'; ~'
If a n a r b i t r a r y
is t a k e n
(3.10)
~ t't' Tt'
~ tt' t~'
the necessary
from a map
~'
t'algebra
We monad
v is an a c t i o n
then
satisfies say that
of monads,
~ = vf'
(3.10);
t in K, the n o t i o n
the
by
is an a c t i o n
o f t o n u' by
is a n a c t i o n
whence
K admits
u'~'
(3.9)
o f t o n u'f' T is a m a p
construction
of t  a l g e b r a
(3.6).
I f v is a n = t',
which
of monads.
of a l ~ e b r a s
c a n be
o f t' on
if,
E
for
"internalized"
in
92
the sense that the 2functor A ~ representable,
Alg(A,t)
from K °p to CAT is
so that
Alg(A,t)
(3.13)
~ K(A,B t)
(2naturally in A) for some B t in K, called the object of talgebras. When this is so, the a d j u n c t i o n of P r o p o s i t i o n 3.1 becomes an adjunction K(A,B t) ~ K(A,B) which, because of its 2naturality in A, arises from an a d j u n c t i o n Bt,ct:
ft~ ut: B t ~ B.
As the n o t a t i o n
CAT does admit the c o n s t r u c t i o n of algebras, and in that
suggests,
case the Bt,ft,u t are those at the end of §3.1. W h e n K = VCAT for a symmetric m o n o i d a l closed category
V,
K admits the c o n s t r u c t i o n of algebras p r o v i d e d that V has equalizers (at least of pairs with a common left inverse); in detail by Dubuc [5].
this case is treated
Here again the primary m e a n i n g of t  a l g e b r a
is one w i t h domain ~, which now denotes not the unit category but the unit
Vcategory;
so in this sense a talgebra is an object s of the
Vcategory B with an action of t on it.
The category A/g(~,t)
of
these talgebras admits a canonical enrichment to a Vcategory, which is B t.
In particular,
taking V = CAT, this applies to the case
K = 2CAT. The best general result we know o f   i t a more general result is c o n t a i n e d in Gray [12] the c o n s t r u c t i o n of algebras 2cate$ory.
is easy to prove and  is that K admits
if it is finitely complete as a
In accordance with the general d e f i n i t i o n of complete
ness for Vcategories in [4], this means that K has all finite limits, that these are p r e s e r v e d by the r e p r e s e n t a b l e and that K admits cotensor products [X,B] and each B in K.
functors K(A,):
K ~ CA~
for each finite category X
It turns out to be sufficient to demand the exist
ence of the C o t e n s o r product [~,B]
where ~ is the a r r o w category
0 ~ i; the existence of the other [X,B]
then follows.
If we replace
"all finite limits" above by "all pullbacks" we get the representable.
93
2categories
of Gray [ii]
so a finitely termi n a l
complete
object
~.4
2category
preserved
We h e n c e f o r t h
algebras.
or of Street's
suppose
morphism
ut~,
check that
and sending
the
ft: B ~ B t, and that
p: A ~ B t r e n d e r i n ~
volume;
one with a
3,.3.
Let
the c o n s t r u c t i o n there
of
is a t  a l g e b r a
an arrow p: A ~ B t to the
2cell
~: p ~ p' to the a l g e b r a
K(A,B t) ~ Agg(A,t).
t: B ~ B arises
the m o n a d
Prooosition
K admits
as follows:
sending
is an i s o m o r p h i s m
the t  a l g e b r a
is a r e p r e s e n t a b l e
that
(3.13)
ut: B t ~ B; and the functor utp,
in this
by the K(A,).
We can express
talgebra
paper [25]
thus
utf t g e n e r a t e d f'~u':
It is easy to
from the a r r o w by ft__~ u t is t itself.
A + B.
Then
the arrows
commutative
(3.14)
P
A
Bt
k/ B
are in b i j e c t i o n with monadmapsT:
with
tactions
t ~ u'f'.
If the monad called the c a n o n i c a l be monadic In fact
u'f'
it is easily
and does not d e p e n d
p: B t' ~
arrow;
the a d j u n c t i o n
if p is an i s o m o r p h i s m
on the choice
B t renderin$
3.4.
f'~
3.2
is a p r o p e r t y
of the left adjoint
There
commutative
ft'
u
t'
is
~' is said to
(an equivalence). of u' itself,
f' to u'
3.3 we let t' be a second m o n a d
{u' to be the a d j u n c t i o n Proposition
by P r o p o s i t i o n
is t itself and ~ : i, the p in (3.14)
seen that m o n a d i c i t y
If in P r o p o s i t i o n take f'
and hence
D
comparison
(weakly monadic)
on u',
on B and
: B t' ~ B, we get:
is a bSject.i.on b e t w e e n
arrows
g4
B t'
P
~
Bt
B and
monad
maps
T:
t
We write it an algebraic clear
that
~
t'.
B T for the p in
map b e t w e e n
(3.15)
the d ~ l
category
K0JB
of objects
provides
a reflexion
duality. particular 3.~ monad
This
For
details
further
in [18]
in 2CAT on CAT.
It is
the canonical
t
t~u
on B fully comparison
of those
,
in t h e arrow
u in K0/B w h i c h
form of the " s e m a n t i c s  s t r u c t u r e "
on the general
laws,
that
we refer
used the name
theory
of monads,
again to Street
"equational
[23].
doctrine"
Here we use the name d o c t r i n e
and in
for a
(or 2monad)
for
in 2CAT. For us,
D: K ~ K with,
then,
a doctrine
for its unit
on a 2category
K
j: I ~ D and m: D 2 ~ D, s a t i s f y i n g
equations
(with D , j , m for t,~,~).
(3.1) Various
to be doctrines by isomorphisms. isomorphisms
relaxations (of.
[15]
What
are possible.
Some
2natural on the nose
have
the equalities
Z~berlein
[27]
calls
in (3.1),
The lax monads
D only
a lax 2functor
and mere m o r p h i s m s
ations
we stick to the
strict
doctrines,
(3.1) not
j and m only
of Bunge in
in
"doctrines"
but have
[3]
(3.1).
and hope
the
things we should
below)
for equalities
transformations.
is a 2functor
and its m u l t i p l i c a t i o n ,
transformations
natural
to T, and call
B t' and B t.
of monads
subcategory
is one
on d i s t r i b u t i v e
Lawvere
any m o n a d
over B;
adjoints.
objects
3.4 as a s s e r t i n g
of the category
into this
corresponding
(3.7).
Proposition
• ~~B T embeds
left
(3.15)
the algebra
internalizes
We can r e g a r d
have
D
replaced
only have lax
are w e a k e r To a v o i d
that
there
like
still:
complicis a nice
95
coherence theorem that will allow the results in our papers below to be applied at least to the "pseudo" case, We also take D  a l g e b r a here in the strict sense: an object A of K (or more generally a 2functor A with codomain K) with an action n: DA ~ A satisfying
(3.2)
considers lax algebras
(with A,n for s,v).
in [25]
below,
However Street
and defines them there;
Kelly considers them in r e l a t i o n to strict algebras below.
When K = CAT, we also use "Dcategory" For m o r p h i s m s of Dalgebras,
are the usual ones in nature,
and [17]
for "Dalgebra".
on the other hand, the lax ones
as we said in §1.5.
from the n o m e n c l a t u r e of ~3.1, and define, D  m o r p h i s m F: A ~ A' ~to be a pair
in I16]
and
We therefore depart
for Dalgebras A,A', a
(f,f) where f: A ~ A' is an arrow in
K and ~ is a 2cell n (3.16)
DA
Df
~
~
DA'
A
f
n'
f
A'
~
satisfying the axioms (3.17)
n
D2 A  mA~
Df
D2flr
D2A ,
D2A
~ A
f
DA'
mA'
n
Dn
DA
f
~
n'
A'
D2f
~ DA
D
D2A '
b
f
f
~ DA' Dn'
A
A' ~
n'
g8
jA (3.18)
n
A
m
DA
=
~ A
identity.
> Df
A'
~
f
DA' ........
jA'
In the the when
case
n'
K : C A Y we a l s o
Dmorphism
F strong
~ = i, so that
fn = n ' . D f .
of D  a l g e b r a s "
Dmorphism
F : (f,l).
If we r e v e r s e and
F = (f,~)
(3.18),
"Dfunctor"
of
the
we
The
§3.1;
sense
get w h a t
is a s t r o n g
for " D  m o r p h i s m " .
if ~ is a n i s o m o r p h i s m ;
"morphisms
(3.17)
say
strict
we a l s o
of ~ in
we
call
Dmorphisms, write
(3.16),
also
we
call
an opDmorphism.
Dmorphism,
then
(f,~l)
call
it s t r i c t then,
f for the
and
We
are
the
strict
in the
axioms
Clearly
if
is a s t r o n g
opDmorphism. Dalgebras of vertical with
the
pasting
same
We n o w m a k e For
and
of diagrams
objects
these
Dmorphisms
~:
into
f ~
like
by restricting
Dmorphisms
be a 2  c e l l
form a category (3.16).
W e get
to s t r o n g
or to
Df
F,G:
DA
A ~ A' we d e f i n e
~A
I
operation
subcategories strict
Dmorphisms.
a D2cell
~:
F ~
G to
g satisfying
DA
Dc~
the
2categories.
n (3 .i9)
under
~
DA
n
~A
b
Df'
f,
~
n'
A v
DA '
'~ A' n~
97
In the case
K = CAT we also
"D2cell".
With
Dmorphisms,
and D2cells
write
as DCAT
(here
the obvious
monadic
sub2category
it is this
of algebras)
that it is doctrinal,
or 2monadic,
in 2CAT, and not just
substantive
are e x a m i n e d
relations
Examples and m o n o i d a l
functors
symmetric
monoidal
categories,
categories arbitrary
and m o n a d
natural
structures
@ and @
appropriate
is the object
Dmorphisms
finite
that we m e a n
some
Dmorphisms
natural
bearing
coproducts,
With
categories
functors,
a monad,
with
bearing
and m o n o i 
with
of Street
arbitrary
categories
functors,
transformations;
monoidal
transformations
2cells.
monoidal
strict m o n o i d a l
and having a d i s t r i b u t i v i t y and
We say of K D
and strict
categories,
categories
transformations;
morphisms
strict
of a l g e b r a s
to go beyond definitions:
symmetric
functor
with a s s i g n e d
only the
sense of §3.3.
and m o n o i d a l
transformations;
functors
here
transformations;
monoidal
monad
in w h i c h
over K (to e m p h a s i s e
monoidal
arbitrary
dal natural
K = Cat.
when
in [16].
of DCAT are:
natural
w h i c h we also
in CAT).
between
b e l o w by Kelly
that
K D in the
It is not our i n t e n t i o n more
DAlg,
or as DCat
K = CAT
are considered;
for
laws of composltion, Dalgebras,
by D  A I ~ the
2category
transformation"
form a 2category
in the case
We denote Dmorphisms
say "Dnatural
the
/23];
functors,
and
two m o n o i d a l
of @ over ~, with
K = CAT 2 the objects
of
DAlg may be pairs of m o n o i d a l categories with a m o n o i d a l functor between
them;
with
K = CAT IAI
for a category
A, the objects
of
DAlg may be lax functors A ~ CAT, and the m o r p h i s m s lax n a t u r a l transformations
(cf. Street
On the other hand categories,
CAT.
the category
and of m o r p h i s m s
nose  the i n t e r n a l  h o m Indeed
be no n a t u r a l
~241).
preserving
monoidal
all the structure
closed
on the
as well as @ and I  is not d o c t r i n a l
it is only a category, definition
of symmetric
of 2cell.
not a 2category: It is monadic
there
over
over
seems
to
CAT, but the
98
monad
CAT is o n l y a f u n c t o r ,
on
3.6
Because
DAlg §3.4
= K D,
some
new
doctrine
sense: acted
questions
objects
of
K acted
D.
Write
U:
DAlg
true
in the
o n by
we
formation
DU
functor
~Jvv~+ U.
G = UE:
sense
that
G: A ~
K with
DAlg
have
about
ors A
~
the
as w e l l
matter
through
the
action
DAlg
the
.
n and
2functor
A ~
well
if we
want
to
need
this
K,
do
stay
as
The 3.4.
A map
gives
a
f in
and
not
D acts
with
E:
an
in
as §3.2
in the
and
only
2functors
oplax;
the
when
certainly
to
laxity
the
in our
papers
K Then
is
not
corresponds
course
DA/~
not
thing an
G only
to
that
a
to a in the
to
and
when we
A
the
do
object
Those 2funct
factorize is
to
of K b u t
definitions
ones
can,
there
of D on
right
given
as we
on U:
of
the
2functor.
oplaxnatural.
A ~
A is
A ~
oplaxnaturaltrans
an a c t i o n G is
primary
2functors
A ~ DAlg
of D c o r r e s p o n d
(3.16),
generality
and
rather
therefore,
reduce
D,
that
but
~vx~
to t h o s e
far f r o m
extra
further
actions
to be
so,
§3.5
together
Perhaps,
the
of
DG
Dalgebras
~ K f o r the f o r g e t f u l
So a 2  f u n c t o r
K
D Alg ,, or
the
o n by
D U ~ U,
A ~
honest
to d e n o t e
sense
transformation
no
2functor.
arise
DAlg
2natural
both
case
using
are
actual
is n o t
weak
a
above. We
it
in the
not
work = ~;
don't
below.
So we
shall
course,
applies
a
equally
but
think
allow
we
we shall
pursue
it
here. same d:
kind
D ~
2functor
D'
of
observation,
of d o c t r i n e s ,
d Alg
of
in t h e
sense
of § 3 . 2 ,
= K ~ satisfying dAlg,
(3.20)
D'Alg,
~
DAlg,
to
Proposition
not
only
99
but also an evident
dAlg s a t i s f y i n g
2functor
dAlg D'Alg
(3.21)
~ DAlg
K
where
of course
sense.
While,
the dA/g,
we are u s i n g U in an e x t e n d e d however,
the only P: D'A/g,
for a 2natural
d: D ~ D' that
P: D'Alg ~ DAlg
many more
oplaxnatural
d: D ~
Again we hope
to avoid
D'
dAlg,
is 2complete, 21imits
but
: K
products.
~20J,
is so if the category
K : Cat/A, show that
K 0 of K is locally
etc.)
for a locally
for example, a left
to
this
from P r o p o s i t i o n
1.5.1
4.1
if the of Manes
of r e f l e x i v e
is so when the under
K ~ Cat, K : C a ~ ,
(e.g.
in their
sense
K 0 the various
We have just not
dAlg has a left adjoint
If K
not only
has c o e q u a l i z e r s
presentable
presentable
coincide.
preserves
cases.
has a 2  1 e f t  a d j o i n t
91,
and when D' has a rank
of whether
are
the algebraic
By P r o p o s i t i o n
D'A/g,
10.3 of G a b r i e l  U l m e r
ions of h a v i n g a rank question
2functor
dAlg,
has a left adjoint.
category
there
axioms.
in suitable
It follows
functor
lying
map,
correspond
that
D' , and dA/g,
underlying
By Satz
with UP : U' are
the a p p r o p r i a t e
we recall
the
pairs.
they
has a left adjoint
so is D'A/g, cotensor
level,
of Kelly I 14] that
this
~ DA/g,
as a r e s t r i c t e d
these.
in (3.20)
also
satisfying
as well
is a d o c t r i n e
with UP ~ U';
While we are at this 2functor
,
 moreover
reasonable thought
in this
they
definit
out the
situation;
jAlg: DAlg ~ lAlg is just U: DAlg ~ K; does it have
adjoint? We have
doctrine
maps
satisfying
in the present
case the n o t i o n
B: d ~ d: D ~ D'; namely
of a m o d i f i c a t i o n
a modification
B: d ~
of
100 (3.22) Here
j'
= ~j, m ' . 6 6
66 d e n o t e s
composite
= 6.m.
neither
the v e r t i c a l
6*6 of m o d i f i c a t i o n s • Dd
(3.23)
~ D~
DD,
~
coincide
frightful
common
D'D'
=
DD
(I.i)
value
and
the h o r i z o n t a l
of D'd
D,D
~6D
~ dD'
by
66 n o r
dD
~6D,
Dd
which
the
dD'
"~___~
DD
but
composite
_~
$ D,6 ~'D, _
~
dD
(1.2);
notation
D'd
in a 3  c a t e g o r y
presents
problems.
If 6 is s u c h a m o d i f i c a t i o n D'algebra n:
with DA
action
n':
of d o c t r i n e
D ' A ~ A, t h e n A has
~ D ' A ~ A, dA n'
It is e a s i l y
verified
(3.24)
DA
that
maps,
dA
if A is a
two D  a l g e b r a
~: DA _~ D ' A ~ dA n' (l,n'.6A)
and
is a D  m o r p h i s m
structures
A.
(A,~)
~
(A,n):
6Alg
rendering
n' ~ A
D'A
6A
DA
~ dA
It f o l l o w s
that
6 induces
~ A
D'A
.
n t
a 2natural
commutative
d,Alg (3.25)
K
transformation
101
in t h e
sense
that
U.~Alg
Proposition
= identity.
3.5.
Any
We
2natural
leave
transformation
D'Alg~Jt~" ~
BAlgf o r It
there
are
a unique that
equivalent
doctrine
maps
d:
D'A/g;
Dmorphism
D'Alg,
and
As ®:
A×A
each
Dalgebra morphisms,
the
prove:
K rendering
m
equivalent.
@
m ) ~ ®,
A when
D'
~ D. = i;
~:
where we
so t h i s
map
let
I: ~ ~ A;
functors n
set
comes A on
go
e:
•
maps,
even
However
to
D'
give
is b y
be
with
= m I +...+
2 @ = @,
m n.
O I = @;
d.e
sense
~ i and
D'Alg** where
no m e a n s
the
the
case
the
stron~ that
DAlg,
A•
with
A with,
for
isomorphisms
A D'algebra
strict
DAlg
equivalent•
category
a category
coherent
e°d ~ i
so t h a t
are
only
a monoidal be
that
2categories
isomorphisms,
and
DAlg
of
a D'category
A n ~ A, a n d
are
maps.
in the
equivalent
DAlg**
it
D,D',
~ D with
here
a Dalgebra
let
m
d ~ d of d o c t r i n e
doctrines
sub2category
are
a D'algebra
e.d
strong•
taken.
morphisms
whence
is
are
strict e:
to
DAlg
6:
modifications
an e x a m p l e •
n ~ 0,
m
gives
is
~ A and
n I @ ( ® ,...,
the
(3.24)
Dmorphisms
D ~ D'
of doctrine
since
DAlg**
where
modification
follows
by modifications and
reader
dAlg
commutative
is
the
morphisms
A gives
a
go
strict
into
from
a d o c t r i n e m a p d: D ~ D'. A Dalgebra 0 1 n n1 s e t t i n g @ = I, ® = i, @ = @(i• ~ ); again
strict
morphisms,
The
composite
DAgg
~
the
composite
D'Alg ~
so t ~ i s
comes
D'Alg ~ DAlg DAgg
~
D'Alg
from
is is
the
A
a doctrine identity,
clearly
102 isomorphic However
to i, whence
the composite
to i; the original
d.e
~ i, and D and D' are equivalent.
D'A/g, ~ DAlg,
~ D'Alg,
and final D'algebra
i: A ~ A is a strong, but not a strict,
is not isomorphic
structures on A are such that isomorphism between them.
~,~IOGRAPHZ [ 1]
J. Benabou, Math.
[2]
Introduction
47 (1967),
M.C. Bunge,
13]
M.C. Bunge,
195 (1971), Coherent
(preprint,
[4]
Dubuc,
C. Ehresmann,
pairs,
Lecture Notes
70122. and relational
Enriched
106 (1969),
Kan extensions
Lecture Notes
[6]
induced adjoint
extensions
B.J. Day and G.M. Kelly,
E.J.
in
algebras
March 1973).
Notes in Math.
[5]
Lecture Notes
177.
Bifibration
in Math.
to bicategories,
in Math.
Cat6gories
80 (1963),
functor categories,
178191.
in enriched
category
theory,
=
145 (1970).
structur6es,
Ann.
Sci. Ecole Norm.
Sup.
349425.
[7]
C. E h r e s m a n n , Cat6$ories ' et structures
[8]
S. Eilenberg
and G.M. Kelly,
Categorical
Lecture
Al~ebra(La
(Dunod,
Paris,
Closed categories,
Jolla 1965.)
Proc.
(Springer,
1965). Conf.
on
New York,
1966). [9]
P. Gabriel and F. Ulmer,
Lokal pr~sentierbare
Lecture Notes in Math.
[ io]
J.W.
Gray, The categorical in Math.
[ 11]
J.W.
99 (1969),
Gray, Report in Z~rich, 195 (1971),
[ 12]
J.W.
=
221 (1971). comprehension
scheme,
Lecture Notes
242312.
"The meeting of the Midwest Category Seminar
August
2430,
1970",
Lecture Notes in Math.
248255.
Gray, Formal category in Math.
Kategorien,
theory, to appear in Lecture Notes
103
[ 13]
A. Grothendieck,
Cat6gories
fibr6es et descente,
SSminaire
de
[email protected]@trie
[email protected] , Institut des Hautes Etudes Scientifiques,
[ 14]
G.M. Kelly,
Paris
Adjunction
in Math.
(1961).
for enriched categories,
106 (1969),
166177.
[ 15]
G.M. Kelly,
On clubs and doctrines
[ 16]
G.M. Kelly,
Coherence
distributive
[ 17]
G.M. Kelly,
[ 18]
F.W. Lawvere,
F.E.J.
E. Manes,
(in this volume).
Ordinal sums and equational
Linton,
and for
(in this volume).
80 (1969),
Autonomous
Journal of Alsebra
[ 20]
(in this volume).
for lax algebras
Doctrinal adjunction
Notes in Math.
[ 19]
theorems
laws
Lecture Notes
2 (1965),
algebras over a triple
Lecture
141155.
categories
A triple miscellany:
doctrines,
and duality of functors,
315349.
some aspects of the theory of
(Dissertation,
Wesleyan Universit~
1967).
[ 2l]
P.H. Pa!mquist,
The double category
Lecture Notes in Math.
[ 22]
J.E. Roberts,
[ 23]
8 (1968),
R. Street,
2 (1972),
functors,
Journal
181193. J. Pure and Applied
149168.
Two constructions
Topologie
squares,
123153.
of initial
R. Street, The formal theory of monads, Algebra,
[ 24]
195 (1971),
A characterization
of Algebra
of adjoint
et
[email protected] on lax functors,
[email protected] Cahiers de
XIII,
3 (1972),
217264.
[ 25]
R. Street,
Fibrations
and Yoneda's
lemma in a 2category,
(in this volume).
[ 26]
R. Street, Elementary
[ 27]
V. Z6berlein,
cosmoi,
(in this volume).
Doktrinen auf 2Kategorien
Inst. der Univ.
ZUrich,
1973).
(Manuscript,
Math.
.FIBRATION$ AND YON.EDA'S ~EMMA IN A 2CATEGORY by ~o~$ Street
Our purpose is to provide w i t h i n a 2category a conceptual proof of a setfree version of the Yoneda lemma using the theory of f i b r a t i o n s .
In doing so we carry
many d e f i n i t i o n s of category theory i n t o a 2category and prove in t h i s more general s e t t i n g results already f a m i l i a r for CAT. The La J o l l a a r t i c l e s of Lawvere [ 5 ] and Gray [ 2 ] have strongly influenced t h i s work.
Both a r t i c l e s are w r i t t e n in styles which allow easy transfer i n t o a 2
category.
However, they also f r e e l y use the fact that CAT is cartesian closed, a
luxury we do not allow ourselves. The 2category is required to s a t i s f y an elementary completeness condition amounting to the existence of 2pullbacks and comma objects.
This relates the
2category closely to a 2category of category objects in a category.
Such con
siderations appear in ~1 and were considered by Gray [ 3 ] . Fibrations over B appear in ~2 as pseudo algebras f o r a 2monad on the 2category of objects over B. Kock [ 4 ] .
This 2monad is of a special kind distinguished by
We define lax algebras and lax homomorphisms for general 2monads and
provide a l t e r n a t i v e descriptions of pseudo algebras and l a x homomorphisms for the special 2monads.
We are able then to give an equivalent d e f i n i t i o n of f i b r a t i o n
generalizing the s e t t i n g for the Chevalley c r i t e r i o n of Gray [ 2 ] p 56. In order to eliminate the need f o r our 2category to be cartesian closed in the remainder of our work we are led to introduce an extra v a r i a b l e ; we must consider b i f i b r a t i o n s from A to B rather than f i b r a t i o n s over B.
A p a r t i c u l a r class of spans
from A to B, called covering spans, is introduced in ~3.
As with t h e i r analogue in
topology, covering spans are b i f i b r a t i o n s . Furthermore, any arrow of spans between covering spans is a homomorphism.
In the case of CAT, b i f i b r a t i o n s correspond to
categoryvalued functors and the l a s t sentence r e f l e c t s the fact that covering spans correspond to those functors which are discretecategoryvalued; that i s ,
105
setvalued.
With this interpretation of covering spans as setvalued functors, we
see that Corollary 16 is a generalization of the Yoneda lemma of category theory. The concept of Kan extension of functors is one of the most f r u i t f u l concepts of category theory, and the definition just begs translation into a 2category. This has already been used to some extent (see [6] and [ 7 ] ) .
But the Kan extensions
of functors which occur in practice are all pointwise (using the terminology of Dubuc [ I ] ) . Using comma objects we define pointwise extensions in a 2category in @4. Note that, in general, for the 2category VCat, this definition does not agree with Dubuc's; ours is too strong (we hope to remedy this by passing to some related 2category). do not.
For V=Sct and V=2, the definitions do agree; for V=AbGp and V=C~t, they
The closing section gives some applications of the Yoneda lemma and
fibration theory to pointwise extensions i l l u s t r a t i n g their many pleasing properties.
106 11. Representable 2  c a t e g o r i e s . Let
A
denote a c a t e g o r y .
A span from
A
to
B
in
A
i s a diagram
(uo,S,ul):
When no confusion is l i k e l y , span
(uI,S,Uo)
w i t h the span
Let
is a b b r e v i a t e d to (1,A,u)
f : ( u o , S , u 1)
from
, (u~,S',u~)
SPN(A,B)
we a b b r e v i a t e
A
S*.
to
B to
to
S;
Also we i d e n t i f y
B.
then the reverse
an arrow
u:A+ B
An arrow o f spans
is a commutative diagram
denote the category of spans from
When A has p u l l b a c k s , a span from
(u0,S,uz)
(uo,S,u 1)
C have a composite span
A to from
B and t h e i r arrows.
A to
(uo~o,ToS,vl~l)
B and a span
from
A to
(v0,T,v 1)
C where the
f o l l o w i n g square i s a pullback. A
Ul
ToS
>" T
S
>B
Ivo
Ul If
f : S     ~ S'
spans from
is an arrow o f spans from
B to
C then the arrow
A
to
gof:ToS
g:T
13 and ; T'oS'
~ T'
i s an arrow o f
induced on p u l l b a c k s i s an
arrow o f spans, An opspan from
A
to
B
in
A
is a span from
arrows o f opspans are arrows o f diagrams in
Suppose A has pullbacks. following data from A:
A
to
B
in
A°P; however,
A,
A category object
A
in
A
c o n s i s t s of the
107  an object 

Ao;
a span (do,Ai,d I ) arrows of spans
from
Ao to
i:(1,Ao,l)
Ao;
~ (do,Ai,dz),
A
A
c:(dodo,AloAl,dldl)
~ (do,Ai,dl);
such that the f o l l o w i n g diagrams commute 1oi
iol
1oc
AioAI
B
f
r/s
r/s~r)B
dz
#
are equal, then there exists a unique 2cell
U l
S~_~r/s
do
S
~ D
such that
f, = do@, q = d1@. In nonelementary terms,
r/s
is defined by a 2natural isomorphism
K(S,r/s) ~ K(S,r)/K(S,s), where the expression on the r i g h t hand side is the usual comma category of the functors
K ( S , r ) , K(S,s).
The comma object of the i d e n t i t y opspan @A.
I t is defined by a 2natural isomorphism
(1,A,I)
from
A to
A is denoted by
109 2
K(S,~A) and so is the cotensor in exists f o r each object
~
K of the category
If
, 2
with the object
A.
A and when K has 2pullbacks we say that
representable 2category (Gray [ 3 ] Example.
K(S,A)
K
is a
uses "strongly representable").
A has pullbacks then
@comma objects in
When "#A
K = CATIA)
is a representable 2category. / /
K are comma objects in
both representable and oprepresentable,
~
K°p.
In a 2category which is
has a l e f t 2  a d j o i n t
is automatically a 21imit in
#
and any l i m i t
which e x i s t s in
K0
Proposition i.
In a representable 2category each opspan has a comma object.
The formula is
Proof.
K.
r / s : s*otDor.
// In a representable 2category, an i d e n t i t y 2cell
arrow
i:A
~ tA,
I A ~ I~A I
corresponds to an
and the composite 2cell ~Ao~A
~A
tA
o
corresponds to an arrow ~od ~ A ~ A
~Ao~A
c
~>A.
For each arrow
f~ B corresponds to an arrow
~f:~A
f:A
, B, the 2  c e l l
~ ~B.
dl Proposition 2.
In a representable 2category the following results hold.
A, the arrows
(a) For each object object
A
in
f:a
enrich
d0,d1:~A
~A
to a category
K 0.
(b) For each arrow arrow
i,c
f:A
..... ~ B,
the pair of arrows
f,tf
constitute a functorial
~ B. f
A ~ . ~ . _ ~ B,
(c) For each 2cell
f, formation from (d) The assignment
to
f'.
the corresponding arrow
a'A
~ ~B
is a trans
110
f
~f 7~'B
~ f,
f,
defines a 2  f u n c t o r from Proof.
K to
(a) For each o b j e c t
functions f o r the category
CATIKo),
X,
IK(X,A)21~
K(X,A);
so
are the source and t a r g e t
Ko(X,@A)'T Ko(X,A)
t a r g e t f u n c t i o n s f o r a category, f u n c t o r i a l l y s t r u c t u r e o f a category object in
]K(X,A) I
K0.
in
X.
are the source and
So @ A ~ A
c a r r i e s the
I t is r e a d i l y checked t h a t t h i s s t r u c t u r e
agrees with t h a t o f the p r o p o s i t i o n . (b) For each
X, ( K o ( X , f ) , Ko(X,@f) )
K(X,f): K(X,A) (c) S i m i l a r l y , K(X,~): K(X,f)
corresponds to the f u n c t o r
~ K(X,B). Ko(X,~)
corresponds to the natural transformation
~ K(X,f').
(d) What we have shown is that the composite
K
~ CAT(Ko)
>
[Ko°P,CAT]
is the Yoneda embedding, a wellknown 2  f u n c t o r . a
I t f o l l o w s t h a t the f i r s t
arrow is
2functor.//
§2. Lax algebras and f i b r a t i o n s Suppose
D is a 2monad on a 2category
denote the u n i t and m u l t i p l i c a t i o n . an arrow
c:DE
C and l e t
i:i
A lax Dalgebra consists of an object
~ E and 2  c e l l s E iE
D2E ~
cE
Dc
DE
~
E
~ D, c:DD
DE
c in the 2category C such that the composites
c
..... )~ DE
~D E,
111 i
DE ......
> DE c
c
~Dc
(1): D E ~ E (2):
@
c
c
1
CDE
CDE
D3E
(3)
• D2E
D3E
~ D2E
DE c
c
A pseudo Dalgebra is a lax Dalgebra in which
are equal as indicated.
A normalized lax Dalgebra has
isomorphisms.
Dalgebra is a lax Dalgebra with both in
C,
DE with
Kock [ 4] c ~iD
CE:D2E
identities.
~ DE is the free Dalgebra on
are
A
Of course, for any E E.
has distinguished those 2monads D with the property that
in the 2functor 2category c.Di D~,~ I
i d e n t i t y modification modification
~,0
~ an i d e n t i t y 2  c e l l .
~,0
Di D , ~
D2 .
iD isomorphism with inverse
[C,C]
Suppose E is a lax Dalgebra such that
~, and consider the composite
E
~ iDE
On the one hand, On the other hand,
Then the
D corresponds under the adjunction to a
DiE D
with i d e n t i t y counit.
c.Dc D2E ~
E . cc E
el E = (CCEiE)(e.Di E) = e.Di E = c.D~. 01E = (eiDE)(C.DC.1 E) = (Cc)(c.DC.~E).
~ is an
112 So we have the e q u a l i t y
DiE D E ~
Dc. Di E
D2E
Dc ~ DE
c ~E
=
(4) c
c) E
D E ' D E I
I
The next proposition generalizes s l i g h t l y some of Kock's r e s u l t s ;
he considers
the normalized case. Proposition 3.
Suppose
D
is a 2monad with the Kock property and suppose
the 2cell E
DE is an isomorphism (a) ~
with inverse
c ~
~E
satisfying equality (4).
is the counit for an a~junction
Then:
c AiE with unit given by the
composite i
°EC
°2E Dc °E
41E /
iE c. Dc
(b)
the 2cell
D 2 E C ~0 ~ E
iDEi E 2cell
E C I ~
DiE.iE holds;
corresponding under adjunction to the identity
c.c E D2E
is unique with the property that the equality (1)
113 (c)
this 2cell
Proof.
(a)
gives
0
E,C,~
enriches
Let
with the structure of pseudo Dalgebra.
denote the composite 2  c e l l displayed in (a).
~c.c~ = 1.
Since the composite
~EiE
is the i d e n t i t y ,
E q u a l i t y (4)
we also have
iE~.~i E = I. (b)
Let
denote the composite
T
DE
I
~DE
i~Dc u b ~ D2E___r._...~,DE
c dE
E Then the 2  c e l l
@ described in (b) is the composite
DiDE D2E
D3E ID2E
iD
Dc
E • D2E
~
Dc
~ DE
c
• E
c
DE The 2  n a t u r a l i t y
of
i:1
, D implies the e q u a l i t y
which i t e a s i l y f o l l o w s t h a t
% = ~c.
i E~c = D#.DC.iDE,
Using t h i s and the equations
from
CEiDE = i ,
IDEiDE = 1, we deduce the e q u a l i t y (1). To prove uniqueness, suppose e
8
satisfies
(1).
The 2  c e l l corresponding to
under adjunction is the composite
iDEi E
(uni t)iDE i E Di E" i ESi DEi E • ~ DiE.iEC.DC.iDEi E ~ DiE.iEccEiDEIE
DiE. i E ( c o u n i t ) DiE.i E. So ( I ) implies t h a t t h i s composite is independent of composite is the i d e n t i t y , 0 (c)
@.
For one such
so the composite is the i d e n t i t y f o r a l l such
@ the 0.
So
is unique. Clearly
@ is an isomorphism, so i t remains to show t h a t
@ satisfies
(2)
114
and (3).
E q u a l i t y (2) f o l l o w s from the equations
c.Dc. DCE.DiDE.t E = c. Dc.t E ,
• = ~c
arrows by t h e i r r i g h t a d j o i n t s " ,
and (4).
CE.Di E = 1, c.Dc.DCE, tDE.DiE =
By the n a t u r a l i t y
e q u a l i t y (3) holds since i d e n t i t y
of " r e p l a c i n g 2  c e l l s appear
in the squares o f the transformed e q u a l i t y . / /
A lax homomorphism o f l a x Dalgebras from f:E
, E'
E
to
E'
c o n s i s t s of an arrow
and a 2  c e l l c
in
C
~ E
DE. . . . c
~ E'
such t h a t the composites c
DE
~E
(5)
DE
E
I .
D2E
m E i
DE'.=~ E'
cE
,L DE
D2f
D2E
=
,I,
)DE
D2E '
~DE'
I
Def IDf
Dc"~ DE' are equal as i n d i c a t e d . ef
cE
c
(6)
when
DE
DE'
c A l a x homomorphism f
i s an isomorphism, and i s c a l l e d a
is called a
homomorphism
~, E'
c
pseudo homomorphism
when
8f
i s an i d e n t i t y .
115 Proposition 4.
f:E
~ E'
D is a 2monad with the Kock property and suppose
Suppose
is an arrow between pseudo Dalgebras.
l:Df.iE~ i E , . f
which corresponds under adjunction to the identity 2cell unique with the property that equality (5) holds, f
ef:c. Df
Then the 2cell
Furthermore, this
~ fc is
enriches
9f
with the structure of lax homomorphism.
Suppose 9f
Proof.
is as explained in the proposition.
since both the 2cells iE,f
1 ~ iE,f
ciE,f
~ f
correspond to the i d e n t i t y 2cell
under adjunction (recall that
On the other hand, suppose
9f
Equality (5) holds
satisfies
~
(5).
is the counit for Then 9f
c ~ iE,)
corresponds under
adjunction to the composite ~.Df.i E iE,Ofi E iE,f~Df.i E ~ iE,c. Df.i E iE,fci E •iE,f which is independent of
@f by (5);
(6) since both the 2cells i d e n t i t y 2cell 8:c.Dc
so
c. Dc. D2f
9f
Finally,
Of
i
DiE,.iE, f
(recall that
I:iDEiE~* DiE.iE). / /
For convenience we henceforth work in a representable 2category Proposition 5. counit
~
Suppose
and unit
~.
f:A
counit the identity and unit
~g
Using
g:C ........ >, B,
the arrow
is a right adjoint for
B:I
d0~ = u~.qd0 Proof.
is an arrow with a right adjoint
~ B
For any arrow
corresponding to the 2cell
satisfies
fcc E correspond under adjunction to the
D2f. iDE,iE = iDE,.iE, f
~ cc E corresponds to
is unique.
,
~ vd I
,
dl:f/g
v:C ~ C
defined by the equations
d1~ = i.
s f . f q = I, we see that the two composite 2cells
K. u, ~ f/g
with
116 I A
f/g
,
~ C ..... \d \
I
A
~ B
~ f/g v /~' /
f/g
~C dI
~ f/g
f/g
~A
~ C~~*B dI
I are equal;
so there e x i s t s a unique 2  c e l l
also see t h a t
Corollary 6.
6v = I.
So
For any arrow
6
is a u n i t f o r
p:E
ip with
unit the identity.
with
is the identity 2cell
h
~
as asserted.
dI ~
~ B, the arrow
Explicitly,p ip
v
Using
ue.nu = 1, we
with i d e n t i t y c o u n i t . / /
do:p/B
~ E
has a left adjoint
is the unique arrow whose composite
E~B. P
Proof.
Since
I:B
> B has a l e f t a d j o i n t , a dual of the proposition y i e l d s the
result.// Corolla~#
dz:f/B
7.
An a r r o w
f:A
, B Foe a right adjoint if and only if the arrow
~ B has a right adjoint.
In this case there is a right adjoint for
dz
with counit the identity. Proof.
If
d I 4 v
f = dlif
I
dov.
Proposition
we can compose with
i f ~ d o
of C o r o l l a r y 6 to obtain
The converse and the l a s t sentence f o l l o w d i r e c t l y from
5.//
C o r o l l a r y 4 applied to
p = i B gives
i
as l e f t a d j o i n t f o r
.do:@B
, B.
la
The u n i t o f t h i s adjunction is the i d e n t i t y and the c o u n i t
@B,.~@B is the 1 2  c e l l defined by the equations d o t o = I , d l t o = ~. D u a l l y , d1:@B , B has i I as r i g h t a d j o i n t w i t h c o u n i t the i d e n t i t y and u n i t t B ~ @ B defined by v
d o t l = h, d l t I = 1.
idl Using the 2pullback property o f the square A
dl
@B
J, B
dl
,
117 loi
we see t h a t
d l t o = 1 = do~ I
i m p l y the e x i s t e n c e o f a u n i q u e 2  c e l l
@B . ~ @ B o ~ B iol
A
such that
A
dol = t o , dlt = tl,
Proposition 8.
(a)
The composite 2cell loi
~B ~
c ~, L
L(E,p) = (tBop,dlp , Lf = 1of, L~ = io~.
denote the 2natural transformations w i t h iol E
The diagrams which say t h a t say t h a t
L
is a 2monad on
Proposition 8 shows t h a t
L
Let
i:l
L,
(E,p)components col
~¢Bop ,
~Bo~BoE
~ ~BoE .
(see Proposition 2 ( a ) ) is a category o b j e c t p r e c i s e l y KB with u n i t
i
and m u l t i p l i c a t i o n
c.
Moreover,
has the Kock property so that Propositions 3 and 4
apply. An arrow
p:E
~B
is c a l l e d a Ofibration over
the s t r u c t u r e of pseudo Lalgebra.
B when (E,p)
The O  f i b r a t i o n is c a l l e d
supports
split when (E,p)
supports the s t r u c t u r e of an Lalgebra. Proposition 9. over
B
(Chevalley criterion].
if and only if the arrow
The arrow
p : ~ E ....~ p / B
p:E
> B is a Ofibration
corresponding
to the 2cell
119
Pdl
OE
:, B
doI E
>B P
has a l e f t a ~ ' o i n t w i t h u n i t an isomorphism.
Suppose
Proof.
(E,p)
The c o u n i t of C o r o l l a r y 6 is
is a pseudo Lalgebra.
r e a d i l y seen to be (id0)ol p/B = ~BoE ~ o o 1 ~
~BoE = p/B
1 t h i s 2  c e l l corresponds to an arrow p/B
k , {(p/B) i
p/Bq~.~p/B
~c
~ {E.
k:p/B
~ ~(p/B).
One r e a d i l y v e r i f i e s
denote the 2  c e l l
L( ;
that
Let
£
be the composite
p£ = k(C%E).
i t is an isomorphism.
Let
~E.~__~E 1
Let
~z denote the unique 2  c e l l s a t i s f y i n g pi = iE). Also
By applying
do(e£.£q) = 1
counit
c
and u n i t
do,d I
(Note t h a t
dos = ~ d 0 , d i s : ( ~ d 1 ) ( c p 1 1 ) .
to
pe.qp
is immediate.
i t is r e a d i l y seen t h a t
lq
To complete the proof t h a t
q, we must show t h a t
d l ( s Z . £ n ) = I.
pe.qp = I.
~ with
dz(sl.l~)
But
=
(TdlZ)(cplzZ)(cL~). From the c a l c u l a t i o n s A
doP11£
= do11£
= ~l = ~.~c.k
= c~k
= C(~ool)
= C(dool)(1ol)
= do(loc)(1ol),
diPli£ = ~p.dl~i£ = l~(pc)k = Idi k = Idi = di(liol ) = di(dlol)(lol) we deduce t h a t
p l i £ = ( l o c ) ( ~ o l ) = Lc.1E.
= di(loc)(iol),
So, by c o n d i t i o n (4), we have
di(eZ.Zn) = (~c)(c. LC. IE)(CL~) = 1. Conversely, suppose Z~p
with counit
£~p
~ and
dlei:d1£Pi dz ~ @E ~ E and
with counit
di [
with c o u n i t
~ d i i = 1.
p/B
~= di~i.
i
~ and isomorphism u n i t
w i t h c o u n i t 1, we have So put
c
di£ ~
n.
Since
pi = i E
equal to the composite
I t is r e a d i l y checked t h a t the composite
120
p/B
E
i
~
B E ~
~ . ~
BE
1
is an isomorphism w i t h inverse the composite 1
Z
E
~ > BE
p/B
~
p/B
~ E
BE
So ~ So
is an isomorphism.
,
> BE
,
1
The e q u a l i t y (4) f o r Proposition 3 f o l l o w s e a s i l y .
E is a pseudo L  a l g e b r a . / / Compare the above p r o p o s i t i o n with Gray [ 2 ] p.56;
so we have r e l a t e d the
d e f i n i t i o n of O  f i b r a t i o n here with the d e f i n i t i o n of o p f i b r a t i o n in K = Cat.
Notice t h a t the u n i t of the adjunction
an isomorphism but an i d e n t i t y . we w i l l
1 IP
[2]
when
f o r Gray is not j u s t
I t is worth poiHting out the reason f o r t h i s since
need the observation in the next paper.
A Ofibration will
be c a l l e d
normal when there is a normalized pseudo Lalgebra s t r u c t u r e on i t .
In Cat every
O  f i b r a t i o n is normal, but in other 2categories t h i s need not be the case. the proof of the Chevalley c r i t e r i o n ,
for a normal Ofibra~ion,
For any arrow
g:B'
p:BE
if
, p/B
~
is an i d e n t i t y then so is
q.
In So,
h a a l e f t a d j o i ~ t with u~it an ide~J~Cty.
, B, " p u l l i n g back along
g"
is a 2  f u n c t o r g * : K ~
KB,;
121 f o r each
E in
KB, the diagram A
g
g*E
B' g
is a pullback.
The c o m p o s i t e 2  c e l l
~/B'
dl B ~
°°I g*E
A
)
BI
g
E
"
P V
induces an arrow of spans gE: A
p/B',
~/
p~;~\
t/o \',, E
B' V
f o r each
E in
KB.
One r e a d i l y checks t h a t
gE' E e KB are the components of a
V
2natural
transformation
g:
KB
g*
> KB,
t st KB
g*
~ KB, v
Indeed, in the language o f Street [ 6 ] ,
the pair
(g*,g)
is a monad functor from
122 to
(KB,L)
(KB,,L)
Proposition 10.
in the 2category 2CAT. g:B' ........> B
Suppose
K.
is an arrow in
For each lax Lalgebra
E, the arrow V
gE
Lg*E enriches
f:E
g*E
~ E'
g*(c)
~ g*LE
g*E
>
with the structure of lax Lalgebra.
For each lax homomorphism
of lax Lalgebras, the 2cell V
Lg*E
enriches
gE
~ g*LE
g*(c)
Lg*E' v ~g*LE' ~ g*E' gE' g*(c) ~ g*E' with the structure of lax homomorphism.
g*(f):g*E
pseudo Lalgebra or an Lalgebra then so is or a homomorphism then so is Corollary 11.
~ g*E
g*E.
If
f
If
E
is a
is a pseudo homomorphism
g*(f).//
The pullback of a (split) Ofibration along any arrow is a (split)
Ofibration.//
Let
R:KA
~ KA denote the 2functor given by:
f (E,q)ZIC(E'
A/f ,q')
'
(A/q'd0)
~
g
A/g
There is a 2monad structure on K by
Kc°.
An arrow
(A/q',d0).
q:E
R and the theory develops as for
>A
is called a 1fibration over
A
L;
j u s t replace
when (E,q)
supports the structure of pseudo Ralgebra. Note that the category
SPN(A,B)
of spans from
2category by taking as 2  c e l l s the 2  c e l l s
a
of
A to
B becomes a
K as in the diagram
123
where
q'o = lq, p'o = lp.
Let
, SPN(A,B)
M:SPN(A,B)
denote the 2  f u n c t o r
given by:
f
1ofol
E ~ E '
1
~ @BoEo@A
 @BoE'o@A.
£io~oI
logol
g
This 2  f u n c t o r supports the s t r u c t u r e of 2monad t o o ; multiplication
c:MM
i:1
~ M and
~ M have as components
ioloi
coloc
E
~ ~BoEo~A and
A span
the u n i t
(q,E,p)
for
A
~Bo@BoEo~Ao~A
to
B
, ~BoEo~A .
i s c a l l e d a bifibration from
supports the s t r u c t u r e o f pseudo Malgebra.
A
to
B when i t
A split bifibration is an Malgebra.
Results on Lalgebras and Ralgebras can be t r a n s f e r r e d to Malgebras via the following result.
The corresponding statement f o r l a x algebras is l e f t
to the
reader.
Proposition I 2. c:¢BoEo~A CL:tBoE
,Suppose
~E
is a span from
A
to
B.
The Malgebra structures
are in bijective correspondence with pairs of arrows of spans
~ E,
structures on
E
CR:EotA E
, E
c L, c R are Lalgebra, Ralgebra
such that
related by the condition that ME
loc R ~
RE
~ LE
I
~ E cR
commutes;
the bijection is determined by loloi
c L = (~BoE
c
~ ~BoEo~A iolol
c R = (Eo~A
c
~ E) c
~ @BoEo@A
~ E)
= CL(lOC R) = CR(CLOl).
Furthermore, an arrow of spans is a homomorphism of Malgebras if and only if it is a homomorphism of both the corresponding Lalgebras and the corresponding Ralgebras.l/
124 Combining this with Corollary I I and the dual for 1  f i b r a t i o n s we have: qorollary 13.
For any arrows
E from
B
A to
f:A'
~ A,
g:B'
~ B, each (split) bifibration g*oEof
induces a (split) bifibration
A'
from
B'.//
to
There is a more general composition of b i f i b r a t i o n s which we w i l l not need. If
E is a b i f i b r a t i o n from
the b i f i b r a t i o n
F ~ E from
A to
B and
A
C can be defined by the usual "tensor product
to
F a b i f i b r a t i o n from
B to
C then
of bimodules" coequalizer, provided this coequalizer exists and is preserved by certain pullbacks. ..k..3. Yoneda's Lemma within, a 2category. Again we work in a representable 2category
K.
A covering span is defined to be a span which is the comma object of some opspan. Theorem 14.
Any covering span is a split bifibration.
Any arrow of spans
between covering spans is a homomorphism. Proof. of
Any comma object
r/s
is a composite
M at the i d e n t i t y span of
D;
so
s*o~Dor.
But
tD
is the value
@D is a free s p l i t b i f i b r a t i o n .
So r/s
is a s p l i t b i f i b r a t i o n by Corollary 13. Suppose f : r / s that
f
~ u/v
is an arrow of spans from
commutes with the Malgebra structures on
12, i t suffices to show that structures separately.
f
r/s
A to
B.
and
u/v.
We must prove By Proposition
commutes with the Lalgebra and Ralgebra
By d u a l i t y , i t suffices to show that
f
commutes with
j u s t the Lalgebra structures. The Lalgebra structure
c:@Bo(r/s)
~ r/s
comes from that of
@D via the
commutative square ~so~ @Bo(r/s) . . . . . . . .
@Do@D
Ic r/s Equivalently, note that from
A to
............
~ ~D
@Bo(r/s)
~B (composed with
is the comma object of the opspan (r,D,sdo)
d1:~B
~ B) since we have the pullback
125 ^
dl .~B
r/sdo
i,o
A
do r/s and
c:r/sd o
~ r/s
~B
dl
corresponds to the composite 2  c e l l r/sd o
^
do The main t r i c k of the proof i s to introduce the 2  c e l l
r/sdo~
A
defined by u/v.
doa
ldo,
=
The arrow
k(f)
dl~
r/s c
= Xdz;
of c o u r s e , we a l s o have such an
for
is defined by the commutative diagram A
A
do
d1
r/s
u/v ~
r/sd o
^ do
u/vd 0
~ ~B
^ dl
~ ~B
The c a l c u l a t i o n s
doaL(f ) = ldoL(f) = 1do = doa = dof~ A
d l a L ( f ) = XdzL(f )
= Xd I
= dl~
=
dlf~
show t h a t the following composites are equal A
A
do r / s d 0 ~
f
L(f)
r/s
~u/v
=
do ulvd0T~.~..~ C
~ulv
r/sdo
C
SO c . L ( f ) = f c , which proves that Let
COV(A,B) denote the f u l l
covering spans. on SPN(A,B);
Let
SPL(A,B)
f
•
is a homomorphism.//
subcategory
SPN(A,B)
whose objects are the
denote the category of algebras f o r the monad M
i t is the category of s p l i t b i f i b r a t i o n s from
A to
B and t h e i r
126
homomorphisms (up to e q u i v a l e n c e ) .
Corollary 1 5.
SPL(A,B)
underlying functor
A
to
B.
~ %PN(A,B)
factors through the
~ SPW(A,B).//
Corollary 16 (Yoneda len~na). span from
COV(A,B)
The inclusion functor
Suppose
f :A
is an arrow and
~ B
Composition with the arrow of spans
bijection between arrows of spans from
to
f/B
E
if:f
F
is a covering yields a
> f/B
and arrows of spans from
f
to
E. Note t h a t
Proof.
f/B
This gives a b i j e c t i o n
is the f r e e Malgebra on the span between arrows o f spans
But by Theorem 14, any arrow of spans Take
Remark.
to
B
K = CAT and
functorially
An arrow
f: ~
>B
is j u s t an o b j e c t
eb
b
, E
A
to
e
from
of
B.
Covering spans
The functor
K(A,B) °p
g A~~ . . ~ B
B
i n t o some category o f sets.
The b i j e c t i o n o f the c o r o l l a r y B(b,)
>e
I b,
~ SPN(A,B)
given by o/B
i
~
f/B
~ g/B
f
is fully faithful. Proof.
The d e f i n i t i o n
o f comma o b j e c t s gives the b i j e c t i o n f
A
~B h >
A
between 2  c e l l s bijection §4.
o
g
between such
h
f
• g/B
~B
and arrows o f spans
h.
and arrows o f spans
The Yoneda lemma provides the f/A
~ g/B.//
Pointwise extensions. Recall the d e f i n i t i o n
of left
f/B
E from
The f o l l o w i n g special case o f C o r o l l a r y 16 appears in Gray [ 3 ] .
Corollary 17.
B.
is a homomorphism.//
between n a t u r a l t r a n s f o r m a t i o n s
obtained by e v a l u a t i n g a t
from
~ E and homomorphisms
in C o r o l l a r y 16.
correspond to f u n c t o r s
becomes the usual b i j e c t i o n elements o f
A =~
f/B
f
f
extension in a 2  c a t e g o r y (see [ 6 ] ) .
and
> E.
127
There is a bijection between 2cells
Proposition 1 8.
dl
j
j/B
• B
A
>×
>
~ B
f
",,, × /
~
exhibits
k
f obtained by composition with extension of left
f
along
extension o f
Proof.
and arrows o f spans j
f
j/B
The f i r s t
f/k.
Yoneda lemma.
The 2cell
if and only if the corresponding
aZong
fd o
By d e f i n i t i o n
o f spans
j
ij.
~ ~X
~ X, t h e r e are b i j e c t i o n s
f/l
~ fd 0
~ I d I
i
(
~ u*
~ f/A
(
I t is readily checked that
n,e
i f and only i f the corresponding An arrow j:A ju C ~ B , jv
f/A > B/u
m • B/u n )f/A
are a unit and counit for an adjunction f m,n are mutually inverse isomorphisms.//
~ B is said to be fully faithful when, given any 2cell u
there exists a unique 2cell
Cf ~ o ~ A v
such that
•
is the
I u
129 u
composite
j
C..~A v
, B.
I t is r e a d i l y seen t h a t
i f and only i f the arrow of spans
@A
~ j/j
j
is f u l l y f a i t h f u l
corresponding to
jh
is an
isomorphism.
Proposition 22.
If
j:A
~ B is ~lly ~ i t h ~ l
a ~ if the 2cell
J A
~B
f
~
k X
exhibits
k as a pointwise left extension of
f along
j, then
o
is
an
isomorphism.
Proof.
Since
k
i s a pointwise l e f t extension and
, j/j
~A
is an isomorphism,
the composite 2  c e l l
~J/J
X
exhibits
kj
as a l e f t extension of
fd o along
corresponding 2cel I
~A
f• kj
exhibits
f
as a l e f t extension of as a l e f t extension of
For a O  f i b r a t i o n pullback of
b
along
By Proposition 18, the
i A
exhibits
d I.
p:E p.
f f
kj along along
~ B and arrow
1A. 1A. b:G
But also the i d e n t i t y So a
2cell
is an isomorphism.//
B, we denote by
Eb
the
130 A
P
Eb
i G
;l
E
b
~B P
Proposition 23.
Suppose in the diagram
P
E
p
that
is a normal Ofibration.
extension of
f
p
along
The 2cell
o
exhibits
if and only if, for each arrow
A
ob
~ B
A
kb
exhibits
as a left extension of
fb
^ E/b
B~ the 2cell
b:G
A
along
p.
p'
~ p/b
BE
p
is a normal 0  f i b r a t i o n ,
(Chevalley c r i t e r i o n ) . adjoint
as a pointwise left
The f o l l o w i n g square is r e a d i l y seen to be a pullback.
Proof.
Since
k
£'
~ p/B
p
has a l e f t a d j o i n t w i t h u n i t an i d e n t i t y
This property is preserved by pullback:
with u n i t an i d e n t i t y .
The arrow
d1:E/b
~ Eb
so
p'
has a l e f t
has a r i g h t a d j o i n t
A
i~:Eb~ E/b
(dual of C o r o l l a r y 6).
right adjoint
p'i^b.
Let
q
So the composite
denote the u n i t o f t h i s adjunction.
checks the equations A
A
b = doP'i b , pdi£' = d I dIZ' p / b ~
and
Pdoq = h .
A  j' ~
diZ':p/b
Eb
o
f
P
~G
....~ Eb
has a
One r e a d i l y
131 A
So
fd0n
exhibits
fb
as a l e f t
extension of
fd o
along
I t follows
dlZ'.
that the composite 2cell p/b
dl
> G
X exhibits
kb
as a l e f t
extension o f
fd o
A
kb
as a l e f t
extension o f
Proposition 24.
K
that
exhibits
dI
i f and only i f
A
qb exhibits
A
p.//
along
Suppose in the diagram
k
J/g
dl
7, C
A
J
~ B
as a pointwise left extension of
composite 2cell exhibits
Proof.
fb
along
T a k e b:G ......~ C.
kg
f
along
as a pointwise left extension of
j. fd 0
Then the along
d I.
The following square is a pullback. j/gb
dz
>G
J/g
dl
~ C
I f this is mounted on the top of the diagram of the proposition we obtain the diagram j/gb
d~
~ G
gb
and this composite 2cell does exhibit
dl
(from the pointwise property of
K).
kgb as a l e f t extension of
fd o along
By Proposition 12 and Theorem 14 we
132 have that
d1:j/g
~ C is a normal O  f i b r a t i o n s (indeed, s p l i t ) .
Proposition 23 applies with
p = dz:j/g~ C to y i e l d the r e s u l t . / /
So
133 Bibl i o~rap.hy. [1]
E.J. Dubuc, Kan extensions in enriched category theory. Lecture Notes in Math. 145 (1970) 1173.
[2]
J.W. Gray, Fibred and cofibred categories. Proc. Conference on Categorical Alg. at La Jolla (Springer, 1966) 2183.
[3]
J.W. Gray, Report on the meeting of the Midwest Category Seminar in Zurich. Lecture Notes in Math. 195 (1971) 248255.
[4]
A. Kock, Monads f o r which structures are a ~ ' o i n t to ~ i t s .
Aarhus University
Preprint Series 35 (197273) 115. [5]
F.W. Lawvere, The category of categories as a foundation for mathematics. Proc. Conference on Categorical Algebra at La Jolla (Springer, 1966) 120.
[6]
R.H. Street, The formal theory o f monads. Journal of Pure and Applied Algebra 2 (1972) 149168.
[7]
R.H. Street, Two constructions on lax funotors. Cahiers de topologie et g6om6trie d i f f 6 r e n t i e l l e XIII (1972) 217264.
ELEMENTARY COSMOI I by Ross S~re~,t
The theory of categories enriched in some base closed category V, is couched in settheory; some of the i n t e r e s t i n g results even require a hierarchy of settheories. nature.
Yet there is a sense in which the results themselves are of an elementary I t seems reasonable then to ask which are the essential elementary results
on which the rest of the theory depends.
In unpublished j o i n t work with R. Walters,
an axiom system was developed which amounts to Theorems 6 and 7 of the present paper restated in terms of the representation arrow. of the desired theory.
We were able to deduce a great deal
One model f o r this system is provided by the 2category
VCat of small Venriched categories together with the 2functor P:(VCc~t)o°°P÷VCat given by PA = lAMP,V] (= the Venriched category of Vfunctors from A°p to V) where V is an appropriate small f u l l subcategory of V. In the case V = SET, V = S e t , there is a universal property of the presheaf construction P which is more fundamental than the axioms mentioned above.
With size
considerations aside this universal property amounts t o , f o r each category A, a pseudonatural equivalence between the category of functors from B to PA and the category of covering spans from A to B.
Generalizing to a representable 2category
K, we obtain the d e f i n i t i o n of an elementary precosmos as presented in this paper, the adjective "elementary" is dropped f o r b r e v i t y .
( S t r i c t l y the universal pFoperty
only determines P:K°°°P÷Kas a pseudo f u n c t o r , so we f u r t h e r ask that there should be a choice of P on arrows which makes i t a 2functor.)
A cosmos is a precosmos f o r
which P has a l e f t 2  a d j o i n t . Our use of the word "cosmos" is presumptuous.
To J. Benabou the word means
"bicomplete symmetric monoidal closed category", such categories V being rich enough so that the theory of categories enriched in V develops to a large extent j u s t as the theory of ordinary categories.
I t is not modifying this meaning much
135 to apply the term to the constuction P of Vvalued Venriched presheaves for such a V, together with whatever structure is needed to make P well defined.
However, we
frankly do not know how a cosmos in this sense in general gives an example of an elementary cosmos in the sense of the present paper. The problem amounts to a wellknown one in the theory of Venriched categories concerning the relationship between comma objects and pointwise kan extensions.
I f we naively take K to be the
2category VC~ then the pointwise l e f t kan extensions given by the coend formula (see Dubuc [6]) are not always pointwise l e f t extensions in K in the sense of the previous paper [22]; the comma objects in K are just not right for extension purposes with a general V.
We conjecture that there is some variant of VCo~twhich
is an elementary cosmos and provides f u l l e r information on Venriched categories (see Linton [19] p 228). Despite this degree of ignorance, we believe there is good reason for presenting our work in i t s present form. Although we do have proofs for many of our results (we mention in particular Theorem 35) in the j o i n t work with Walters, the proofs of the present paper are shorter and simpler.
Further, our work can be regarded as a
different approach to the elementary theory of the (2) category of categories emphasising the role of the setvalued presheaf construction (compare Lawvere [14]). Also the (pre) ordered objects in any elementary topos provide an example of an elementary cosmos; in particular, the 2category of orderedsetvalued sheaves on a site is a cosmos. This observation contributes to topos theory in that our theory puts the techniques of adjoint arrows, kan extensions, comma objects, completeness, etc, at our disposal to examine the everpresent ordered objects in a topos.
The
2category of categoryvalued sheaves on a site is most probably a cosmos. Finally, we repeat the hope that enriched categories can be shown to f i t
into our present
framework and mention that in a forthcoming paper we w i l l show that CoJtenriched categories ( t h a t i s , 2categories) do f i t The notations and results of [12] paper.
in by expanding to double categories. and [22] are f r e e l y used throughout this
136
Table of Contents page §1.
Internal attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudo f u n c t o r i a l i t y of
138
SPL; 2categories endowed with a t t r i b u t e s ; admissi
b i l i t y and legitimacy; new characterizations of adjunctions and pointwise l e f t extensions.
§2.
Precosmoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
L43
D e f i n i t i o n ; homstatement of the Yoneda lemma; extension and l i f t i n g
properties
of homarrows.
§3.
The representation arrow Definition of
PgA  t
147
YA ; relation between covering spans and a t t r i b u t e s ; denseness
and f u l l y faithfulness of adjunction
................................
gA; existence of internal r i g h t extension V f ;
the
gPA; the ChevalleyBeck conditions; formula for l e f t extensions
for absolutely cocomplete objects; degeneracy.
§4.
Extension systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
D e f i n i t i o n ; special cases; the extension system Pro~; closed and enriched categories of arrows; tensor product of arrows; the extensional bicategory Pro~(G}.
§5.
Cosmoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
161
The comprehension scheme; existence of internal l e f t extension
3f; definition
of a cosmos; l e f t extensions which are always pointwise; small objects and kan arrows; r i g h t extensions; more on 3f.
~6.
Universal constructions
166
"Colimits" in 2categories; examples including pushouts, opcomma objects, k l e i s l i constructions, l o c a l i z a t i o n s ; relation between eilenbergmoore and k l e i s l i
137 page constructions; recognition of objects of the form
~7.
Examples
..........................................
PK.
I72
Ordered objects in a topos; the preSpanier construction; categories.
138 1.
Internal a t t r i b u t e s . Let
category
K be a representable 2category. SPN(A,B)
of spans from
each span S the span
A to
For objects
B supports a monad M which assigns to
MS obtained as the inverse l i m i t of the diagram ~nA
S
h°mB
where, in t h i s paper, we denote the comma o b j e c t c a l l the Malgebras s p l i t b i f i b r a t i o n s from is s t r i c t l y
A, B, recall that the
A/A = ~A by
A to
homA.
We f r e e l y
B, although a s p l i t b i f i b r a t i o n
the underlying span of an Malgebra.
Before giving the next d e f i n i t i o n we must describe a pseudo functor SPL: K c ° ° p x K °p
For each pair of objects from
A
to
B.
SPL(A,B)
A, B,
is the category of s p l i t b i f i b r a t i o n s
For each pair of arrows
f: A' ~
g oof: SPN(A,B) lifts
~ CAT.
A,
g: B'* B, the functor
~ SPN(A',B')
in a canonical way (see Corollary 13 of the l a s t paper) to a functor
SPL(f,g) = g*oof: SPL(A,B) h For 2  c e l l s
A~.~A,
g B'~T
f
B,
the natural transformation
k
SPL(o,T): SPL(f,g) ~SPL(h,k) with action
~ SPL(A',B').
is defined as follows.
For each
c: ME ...... ~ E, the component SPL(~,~) E = T*oEo~
SPL(A,B)
E in
is the composite
g*oEof !°iE°~ g*oMEof=(B/g)oEo(f/A) (B/T)°I°(~/A)~(B/k)oEo(h/A):k*oMEo,h~°c°~ k*oEoh. The routine v e r i f i c a t i o n s required to prove that
SPL is a pseudo functor are l e f t
to the reader.
P: Kc°°p
Suppose we have a 2functor split bifibration
EA from
a t t r i b u t e s when the functors
A
to
PA.
> K and, f o r each object
A
in
These data are said to endow K with
K, a
139
K(B,PA)
{A,BI}
SPL(A,B)
h B ~ A
I
, (h*o A
k*o A)
k are f u l l y
faithful
and form the components of a pseudo natural transformation.
Pseudo n a t u r a l i t y o f
{A,BI}
in
(Pu)*oeA~ ~A, OU n a t u r a l l y in o f the form We c a l l
{A,Blh}
is automatic, but in
u: A   ~ A '.
f o r some h: B
A
i t means t h a t
Split bifibrations
isomorphic to ones
~ PA are c a l l e d a t t r i b u t e s from
A
to
B.
PA the object of attributes of type A.
A p a i r o f arrows
a: X
to a split bifibration Y.
B
~ A,
E from
Then there is an arrow
b: Y   * B is said to be admissible with respect
A
to
B when b*oEoa
E(a,b): Y
is an a t t r i b u t e from
X
to
~ PX defined uniquely up to isomorphism by
the c o n d i t i o n {X,YIE(a,b)} = b*oEoa . For 2  c e l l s
~: a'
8: b
~ a,
are admissible with respect to
) b',
where both the pairs
E, we can d e f i n e a 2  c e l l
so t h a t the c o n d i t i o n of the l a s t sentence becomes n a t u r a l .
a, b
and
a ' , b'
E(a,~): E(a,b)  ~ Then
"E"
E(a',b')
becomes a
pseudo natural t r a n s f o r m a t i o n in the f o l l o w i n g sense. Proposition 1.
If the pair of arrows
respect to a split bifibration ~ Y, the pair
v: K
au,
bv
E
a: X
from
A
~ A,
to
B
b: Y ~
B
is admissible with
then, for all arrows
is admissible with respect to
E
u: H
~ X,
and there is a
natural isomorphism E(au,bv) i~roof.
~
Pu.E(a,b).v .
The f o l l o w i n g isomorphisms are a l l n a t u r a l : (Pu.E(a,b).v)*oC H ~ v*oE(a,b) o(Pu) oEH ~ v*o(E(a,b) OEx)OU v*o(b*oEoa)ou ~ (bv)*oEo(au
We say t h a t an opspan
X f,
A
~ E(au,bv)*oE H . // 9
Y is admissible when the p a i r
f,g
is
140
hOmA;
a d m i s s i b l e w i t h r e s p e c t to from
X
to
Y.
homA(f,g): Y
or in o t h e r words, when
~ PX
f:
ooadmissiblewhen when
homA
i s an a t t r i b u t e
As a s p e c i a l case o f our above n o t a t i o n , we then have an arrow satisfying
the c o n d i t i o n
{X,YIhomA(f,g) ~ ~ Call an arrow
f/g
×
admissible
~A
the opspan
is an a t t r i b u t e .
1A,f
f/g
.
when the opspan
is admissible.
f,l A
is admissible;
Call an o b j e c t
call
legitimate
A
The f o l l o w i n g i s an immediate consequence o f
P r o p o s i t i o n 21 of the l a s t paper. Theorem 2.
f:
B
An arrow
~A
U: A
~ B is a right adjoint for an admissible arrow
if and only if there is an isomorphism
hOmA(f,l) ~ hOmB(l,u). Recall t h a t an o b j e c t
G is said to be
orthogonal
to an arrow
f: A
~B
when the f u n c t o r K(G,f): is an isomorphism. when i t
gonal to
on o b j e c t s and f u l l y
strongly generating f
then
' K(G,B)
This is an elementary c o n d i t i o n since a f u n c t o r i s an isomorphism
is b i j e c t i v e
said to be
K(G,A)
f
When the arrows
faithful.
when, given
f: A
A class ~ B,
if
G of objects of each
G in
j:
A
> B,
f " A ~X
J
are each a d m i s s i b l e , we have the
X
A )
B
A/I , X
Yoneda
p2A
PA are legitimate then
PA
with counit an isomorphism.
Take f = YA in Theorem 11. ¥YA =
PYA: P2A
By Theorem 8 we have
h°mpA(h°mpA(YA'l)'l)
~ h°mpA(l"l)
:
YPA"
is a left
150 The counit is an isomorphism since Proposition 13.
A f , X Jg B,
YPA is f u l l y f a i t h f u l . ~
For any legitimate object
A and any admissible opspan
there is a 2cell dl
f/g
~"B
~PA
A
YA which exhibits Proof.
homx(f,g )
as a pointwise left extension of
YAdo along
d I.
Consider the composite 2cell dl
yA/ hornx ( f , g )
~'" B
YA
/ ~
A
°mx(f'g) ~ PA
By Theorem 10, the lower t r i a n g l e is a pointwise l e f t extension; 24 of the l a s t paper, the composite exhibits sion of f/g ;
YAd0 along
d I.
homx(f,g )
so the r e s u l t follows.#
Chevalley [ 4 ] .
Rf
as a p o i n t ~ i s e l e f t exten
Using Theorem 8 we have YA/hOmx(f,g) ~ homx(f,g)*oE A
Functors which are simultaneously O and
R: B°p
so, by Proposition
~ CAT ;
1fibrations have been considered by
A Ofibration over a category
B corresponds to a pseudo functor
when the original Ofibration is also a 1  f i b r a t i o n , the functor
has a r i g h t adjoint
and BenabouRoubaud [ 3 ]
v
Rf
for each arrow
f
in
8.
Chevalley (and l a t e r Beck
in t h e i r study of descent data) considered a compatibility
condition on the f i b r a t i o n which, in terms of
R, amounts to the following:
151 k
P
, B
hI
f o r each p u l l b a c k
RA
I0
A
i
~ C
Rh
~'RP
riCO ~ RB
RC
f
Rf
i s an isomorphism. This statement s t i l l
makes sense v e r b a t i m in the case where
the r e s u l t i s too stror~.. t h i s case.
The f i r s t
is to require that
levels!
i s a 2  c a t e g o r y , but
There are two g e n e r a l i z a t i o n s which present themselves in
is a category t h i s i s no c o n d i t i o n on a t two d i f f e r e n t
B
f
should be a O  f i b r a t i o n in
f);
B
(if
B
an i n t e r e s t i n g combination o f f i b r a t i o n s
The second i s to replace the p u l l b a c k by a comma o b j e c t
dl f/g
~ B
A
~C f
in
B (if
B is a category this reduces to the pullback again).
The two generali
zations are closely related and we have both in a precosmos. Theorem 14.
Suppose
f: A
C,
g: B
~ C are arrows between legitimate objecta
Then the 2cell
Pd0 PA
~ P(f/g)
PC . . . . . . . . . . . Pg corresponding under adjunction to
Proof.
Proposition 13 yields that
YAdo along
dI.
PB
Ph, is an isomosphism whenever
homc(f,g)
~f, Yd I
exist,
is a pointwise l e f t extension of
By Theorem 3, we have an isomorphism
~mc(f,g)/PA ~ homB(dl,l)/homPA(yAd o,1). Now hOmpA(YAd0,1)
~
Pd0.~mmpA(YA,1) ~
Pd0 ;
and a l s o
hOmB(d1,1)
i s admissible
152 when Vd I
exists (Theorem 11).
So the span on the r i g h t hand side of the displayed
isomorphism above is an a t t r i b u t e and we have an isomorphism
~mpA(homc(f,g),1) ~ ~mp(f/g)(hOmB(d1,1),Pdo). Using Proposition 1 and Theorem 11, we obtain the result.# Theorem 15.
Suppose
and
are legitimate.
E, B, G
p: E
> B
is a normal Ofibration,
b: G
~ B is an arrow,
Then the 2cell
PE
)" PEb
¥p
p
PB
~
PG
Pb
corresponding under adjunction to the identity 2cell morphism whenever
Proof.
~p, ~
~ P~.Pp, is an iso
exist.
From Proposition 13, the 2  c e l l
l e f t extension of
P~.Pb
YE along
p.
×
exhibits
hOmB(P,1) as a pointwise
By Proposition 23 of the l a s t paper t h i s implies
that the composite A
Eb
~, G
P
E
exhibits
hOmB(P,1)b
>B
as a pointwise l e f t extension of
A
yEb along
A
p.
By Theorem 3,
we have an isomorphism A
A
hOmB(P,b)/PE ~ hOmG(P,l)/hOmpE(YEb,1). But
hOmpE(YE~,l)
~
P~ , and
A
hOmG(P,1)
A
is admissible when Yp
exists.
So the
span on the r i g h t hand side of the above displayed isomorphism is an a t t r i b u t e .
This gives:
153
hompE(hOmB(P,b),l) ~ hOmpEb(hOmG(~,l),~). The r e s u l t follows using Proposition I and Theorem I I . #
Remark.
Consider the case where
K = CAT. The isomorphism of Theorem 14 expresses
i n t e r n a l l y the fact that r i g h t extensions of setvalued presheaves are pointwise. Of course, the arrows can be replaced by t h e i r l e f t adjoints to obtain an isomorphism 3do PA
X
The 2  c e l l e x h i b i t i n g t h i s ex
is f u l l y f a i t h f u l . YA PA
X
f: A
ool@nit
154
Theorem 16. f: A
Suppose along
~X
A
is legitimate.
The pointwise left extension
exists if and only if, for all admissible arrows
YA
the pointwise left extension
k
composite
of
f
along
hOmB(J,1) Since
exists.
~ PA
j: A
In this case, k
~ B,
is the
~ X.
YA is admissible (Theorem 8), " i f " is clear.
By the pointwise property, for any
of
lexf
B
Proof.
j
lexf
g: C
Suppose l e x f
exist~
> B, the composite 2cell dl
YA/hom(j,g)
~' C
YA
A
• PA
X exhibits
lexf .hom(j,g)
kg =
YA/hom(j,g)
span
can be replaced by
Proposition 17. lexf
as the l e f t extension of
Suppose
A
lexf
we should have
f'
If
homx(f,1)
leXA: PA so
If
has a left
~ A.
If
f
by Theorem 3;
along
YA'
and this
is the i d e n t i t y arrow of the l e g i t i m a t e object
then
absolutely cocomplete when i t
= PYK ;
is admissible.
hom(YA,1)/hom(f,1) ~ PA/hom(f,1) condition that f ' 4 hom(f,1).~
deserves special a t t e n t i o n ;
adjoint
homx(f,1).
>X
should be a pointwise l e f t extension of
The p a r t i c u l a r case when f
to be
f: A
f'/X ~
is precisely the
But the
exists.
In order that
Proof.
d 1.
(Theorem8).~
is legitimate and
exists then it is a left adjoint for
adjoint then
leXA
j/g
fd o along
hom(f,1) = YA: A
~ PA.
An object
is l e g i t i m a t e and YA: A
A = PK for some l e g i t i m a t e
A is absolutely cocomplete.
A
A
is said
> PA has a l e f t
K then Theorem 12 gives
For absolutely cocomplete objects we
have the most favourable form of a d j o i n t functor theorem.
155
Theorem 18.
Suppose
A
is absolutely cocomplete.
has a right adjoint if and only if lexf
~ flex A If
Proof.
lexf
~X
f: A
exists are the canonical 2cell
is an isomorphism.
f~
u
then, composing with
u = hom(l,u) ~ h o m ( f , l ) .
flex A ~yA
An admissible arrow
So
YA ' we have
leXA~
lexf
exists and is isomorphic to
f /exA. Suppose f l e x A ~ l e x f .
u = leXA.hom(f,1) f
Theorem 16 applies to
as the l e f t extension of
preserves this extension.
Remark.
1A along
f
along
f
A
is the terminal object in
cocomplete categories have terminal objects. c o l i m i t s in
X is: (lexf)G
=
X is obtained from
f: A
> X by evaluating at the terminal presheaf.
the i d e n t i t y functor of
I t remains to prove that
f.
(Theorem 16). H
For categories, the c o l i m i t of a functor
l e x f : PA
to y i e l d
fu = f leXA.hom(f,l) ~ lexf .hom(f,1), and the
But
l a t t e r is the l e f t extension of
A
1: A
A.
Recall that the c o l i m i t of I t follows that absolutely
The formula f o r
lexf
in terms of
fa
Ga ® f a . #
Since we have discussed the case where the representation arrow has a l e f t a d j o i n t , we digress b r i e f l y to point out some t r i v i a l i t i e s i t has a r i g h t a d j o i n t .
An object
A
regarding the case where
is called degenerate when i t is l e g i t i m a t e
and YA: A ..... ~ PA has a r i g h t a d j o i n t . Proposition 19.
A degenerate object
A has the following properties:
(a)
YA: A+PA
(b)
each admissible arrow with source
(c)
each arrow
some
S: B
Proof. yAt   ~
B ~
RA
A
has a right adjoint;
is isomorphic to one of the form
hOmA(1,s)
for
~ A.
Let I
is an equivalence;
t
be a r i g h t a d j o i n t f o r
is an isomorphism since
YA"
Recall Theorem 10.
YA is dense, and the unit
The counit 1   ~ ty A
is an
156 isomorphism since 2 to obtain that
YA is f u l l y f a i t h f u l . u = t hom(f,1)
This proves (a) and (b). K(B,A) ~ K(B,PA);
§4.
So YA: A ~ PA.
Now we can apply Theorem
is a r i g h t a d j o i n t f o r admissible
f: A
~ B.
Using (a) we have an equivalence of categories
(c) follows from t h i s and Theorem 8.#
Extension s£stems. Monoidal (= m u l t i p l i c a t i v e ) categories [ 1 ] , [7] have been generalized by
Benabou
[ 2]
to bicategories.
We now make the corresponding generalization for
closed categories. An e x t e n s i o n system
E
(i)
objects
(ii)
f o r each pair
consists of the f o l l o w i n g data:
A, B, C . . . .
A,B
;
of objects, a category
called arrows and whose arrows are called 2  c e l l s (iii)
for each object
(iv)
f o r objects
A, an arrow
f o r arrows
whose objects are
;
YA c E(A,A);
A, B, C, a functor [ , ] : E(X,A) °p × E(X,B)
(v)
E(A,B)
f ~ E(X,A),
g c E(X,B),
h Vf,g: [ f , g ] ........ [ [ h , f ] , [ h , g ] ] ,
~ E(A,B) ;
h c E(X,C), 2  c e l l s
f × : YA
~ [f'f]'
mg: g    ~ [ y x , g ]
,
the l a t t e r an isomorphism; such that the following axioms are s a t i s f i e d : ESI.
h f Vf,g, × , ~g
are natural in t h e i r subscripts and extraordinary natural in
t h e i r superscripts; ES2.
the f o l l o w i n g diagrams commute
157 [f,f]
[f,g]
v• [[g,f],[g,f]]
\ / YA'
v ,[[f,f],[f,g]]
[f,g]
v
YX ~ [[yx,f],[yx,g]]
S
[i,~] [YA,[f,h]] h)
[f,g]
[~,i]
[f,[YX,g]]
k [[k,f],[k,g]]
[ l , v g]
[[h,f],[h,g]]
~)[h,k]~
[[k,f],[[h,k],[h,g]]]
[[ [ h , k ] , [ h , f ] ] ,[ [ h , k ] , [ h , g ] ] ]
;
[wh,1]
the composite function
ES3.
[,g]
E(A,A) (×g,1) E(a,A)([g,g],[f,g])
E(X,A) (f,g)
E(A,A) ( y A , [ f , g ] )
is a bijection. Special cases.
An extension system with precisely one object is a closed
1)
category.
This does not quite agree with the d e f i n i t i o n of closed category appeaHng
in [ 7].
Reference to a category of sets has been eliminated as required for exam~e
by Lawvere [ i ~ X @
p12. Also, a monoidal category such that each of the functors
has a r i g h t adjoint is closed in our sense (compare t h i s with [7 ] Theorem
5.8 p493).
Note that, for any extension system
E,
E(A,A)
becomes a closed
category. 2)
A bicategory
system with
[f,h]
B in which a l l r i g h t extensions e x i s t yields an extension
taken as the r i g h t extension of
h along
f.
Such a bicategory
we call an extensional bicategory (also called "closed bicategory" by some authors). 3) k: A
Suppose E is an extension system such that, for a l l > C, there exists an arrow
k ~f:
[k ® f ,  ] Then
D ~
f: D
~ A,
> C and a natural isomorphism [k,[f,]].
E becomes an extensional bicategory with composition given by ®.
158
In a precosmos K, suppose the arrow
h: C
X
is admissible.
The
composite 2cell
d~ f/g
)
B
~,
hom(h,f) induces an arrow of spans f / g
Provided the source and
~ homx(h,f)/homx(h,g).
target of the l a t t e r arrow are a t t r i b u t e s , we obtain a 2cell ~,h:
h°mx(f'g)
....... h ° m p c ( h ° m x ( h ' f ) ' h ° m x ( h ' g ) ) "
The next two theorems can be proved using Theorems 6 and 7. Theorem 20. (i)
An extension system
Prof
is defined by the following data:
the objects are the legitimate objects
A
of
K
PA
for which
is
legitimate; (ii)
Prof(A,B) = K(B,PA); g c Prof(X,B), take
f ~ Prof(X,A),
(iii) f~r (iv)
YA is the representation arrow;
(v)
~, X
Theorem 21. data
For each object
hOmA, w, X
VK is
where
K(K,): L
are as previously defined and of
Prof
enrich the category
Prof(K,K)
~ CAT
K
g © f:
2cell:
is the isomorphism of Theorem 8.#
and each legitimate object
K(K,A)
lifts to a 2functor
L
Given arrows
f: A
~ X,
A
of
K, the
with the structure of a VKcategory,
with its closed category structure.
Indeed, the 2functor
........>' V K  CAT.~
Freyd's tensor product of functors [ 8 ] a precosmos.
~
I f , g ] = k~mx(f,g);
g: B
p120 can be carried over to arrows in > PA,
t h e i r tensor product
B    ~ X , when i t e x i s t s , is defined as the l e f t extension as e x h i b i t e d by a
159
{A,BIg }
, B
t
FI
A
If
h
A f~x
~X
is an admissible opspan then there is a b i j e c t i o n
~C
{A,BIg}"
{A,BIg}
~ B
I FI A
h
~
~
\
~ X
{A,BI }
/
{A,Blhom(f,h)}
Using the l e f t extension property of f a i t h f u l n e s s of
g®f
g ® f
on the l e f t hand side and f u l l y
on the r i g h t , we obtain a natural b i j e c t i o n between
2cells: g®f
g
h Proposition 22. lexf
exists.
Suppose
hom(f,h)
A is legitimate and f : A
Then, for all
g: B~ PA,
g ® f
> X is an arrow for which exists and there is a natural
isomorphism g ® f
Furthermore, if
~
(lexf)g.
X is legitimate then, for all arrows
h: C
Xj there is a
natural isomorphism
homx(g ® f , h ) Proof.
{glA, B} = YA/g ;
By Theorem 8,
pointwise property of
~
lexf.
hOmpA(g,~mx(f,h)). so the f i r s t
isomorphism follows from the
When X is l e g i t i m a t e the l e f t hand side of the
second isomorphism e x i s t s , and
{B,Clhom(g ® f , h ) } so the arrow
~
hom(g ® f , h )
g ®f/h
~
(lexf)g/h
~
g/hom(f,1)h
has the defining property of
~
g/hom(f,h) ;
hom(g,hom(f,h)).#
160 Let
G denote a class of objects of
Gcocomplete when, f o r each exists.
I f each
X G in
is
G in
G in
An o b j e c t
G and each arrow
X of
f: G
K
> X,
is c a l l e d lexf:
PG
~X
G is l e g i t i m a t e Theorem 16 i m p l i e s :
Gcocomplete i f and o n l y i f ,
G and a l l arrows
K.
f: G
f o r a l l admissible arrows
j:
G
~ X, the pointwise l e f t extension o f
f
~ B with along
j
exists.
Theorem 23. for each
Let
G
objects in
in G
G G.
be a class of objects of The restriction
K is in
K(K,A) becomes
Gcocomplete then
of
such that
PG
is Gcocomplete
of the extension system
Prof
to
is an extensional bicategory with composition given by tensor product,
In the notation of Theorem 21, if
Suppose
Prof(G)
Prof
G
A
and
is legitimate and
a tensored VKcategory. ~
G is as in the l a s t theorem o n l y regard i t as a f u l l
K, and l e t
I : Gc°
> Prof(G)
sub2category
denote the pseudo functor which is the i d e n t i t y
on objects and which is given on homcategories by the functors K(A,B)oP the isomorphisms
horn(,1)
l ( g f ) ~ Ig ® I f
,
; K(B,PA)
=
Prof(A,B) ;
IIA = YA are canonical.
The f o l l o w i n g p r o p o s i t i o n extends a theorem o f Benabou on profunctors to our setting.
Proposition 24. Indeed, if Eroof.
Arrows in the image of
f: A
The 2  c e l l
hom(f,l) ®hom(l,f) g: B .... i C in
B
is in
G
then
× ' : hom(f,1)
I
have right adjoints in
Prof(G).
homB(1,f)in
Prof(G).
hOmB(f,1 ) ~
~ hom(hom(1,f),#Dm(1,1))
hom(l,1) : YB"
What we must show is t h a t , f o r a l l arrows
Prof(G), the composite 2  c e l l B
corresponds to a 2cell
~m(1,f)
y~~~om
C
~ A
(f,~)
161 exhibits
g ®hom(f,1)
as a r i g h t extension of
g ®YB
along
are in an extensional bicategory so the r i g h t extension of hom(l,f)
hom(1,f).
g ®YB = g
But we
along
is
hom(hom(l,f)g)
~
Pf.g
~
g ®horn(f,1),
as required.g Lawvere [18] has viewed nonsymmetric metric spaces as enriched categories and found a condition which can be stated in our context and reduces to Cauchy completeness for metric spaces. when each arrow
A+X
arrow of the form
§5.
An object
X of
Prof(G)
in
homx(f,l)
Prof(G)
is called Cauchy(G)complete
with a r i g h t a d j o i n t is isomorphic to an
(compare Proposition 19).
Cosmoi. Another way of expressing the precosmos condition is:
for all objects
A, B,
the composite functor K(B,PA) is f u l l y f a i t h f u l .
{A~BI}
>
SPL(A,B)
und
,
SPN(A,B)
We now show that r e f l e c t i o n s with respect to t h i s " i n c l u s i o n "
functor are j u s t certain l e f t extensions. Theorem 25. and that hOmA(1,u)
(Comprehension scheme).
u: S
along
>
is coadmissible.
A
v
Suppose
(u,S,v)
is a span from
The left extension
k: B
~ PA
of
exists precisely when the left adjoint of the functor
{A,BI}: K(B,PA)
,SPN(A,B)
exists at the span (u,S,v). Proof.
A to B
For any arrow
h: B S
"•,
, PA, there are bijections S
pullback property
{A,BIh}
\
/ V*o{A,BIh}
162
A/u do///
~
V
S
dl
~ B
/
horn (
~
h PA k
So we have the bijection between 2cells
a
and 2cells
B~
i f and only
$T~A h
f
i f we have the b i j e c t i o n between arrows of spans Proposition 26.
Suppose the span
which is an admissible opspan.
(Lax D o o l i t t l e ) .
COV(A,B)
A
from
A
to
B
T.//
has an opcomma object
is legitimate then the left adjoint in the
(u,S,v).
last theorem exists at
}Woof.
If
(u,S,v)
and 2  c e l l s
The r e f l e c t i o n for the inclusion functor
>SPN(A,B) is obtained by forming the comma object of the opcomma object.
When A is l e g i t i m a t e , AZ~t(A,B) c COV(A,B). The condition of the theorem ensures that the r e f l e c t i o n lands in Theorem 27. Pf:
PA
Suppose
.....~. PB
extension of
Proop°.
A~Jt(A,B) ~ K(B,PA).#
A is legitimate and
has a left adjoint hOmB(1,f ) along
By Theorem 5,
3f: PA
> PB
YA
exists.
3f
~
hom(1,f)
f: A
~ B is coadmissible.
The arrow
if and only if the pointwise left
In this case, there is an isomorphism
lexhOmB(1,f ).
is admissible and
hom(hom(1,f),l) ~ Pf.
The
r e s u l t now follows from Proposition 17.// Under the conditions of the l a s t theorem, we have an isomorphism
A
which exhibits case where
3f
YA
~PA
as a pointwise l e f t extension of
hom(l,f)
B is legitimate this is the more f a m i l i a r diagram
along
YA"
(In the
163
I
YA
A
~PA
B
>
PB
YB which expresses the pseudo naturality of
y: i ...........> 3.)
Theorem 25 and Proposition 26 provide conditions under which certain l e f t extensions e x i s t .
I f these results are to be used to produce 3 f , we need to know
that the extensions are pointwise (by Theorem 27).
In a cosmos we shall see t h a t ,
with size conditions, l e f t extensions of arrows into objects of the form necessarily pointwise. (Proposition 26).
PK are
At best t h i s approach requires opcomma objects in
K
In order to continue t h i s approach and to present another approach
which does not depend on opcomma objects (but does s u f f e r from size problems in the nonuniform case) we require
K to be a cosmos.
A cosmos is a precosmos for which the 2functor
adjoint
P*: K
P: Kc ° ° p
>K
has a l e f t 2
This means of course that there is a 2natural isomorph~m
.~. ~ , o o p
K(A,P*B) °p ~ K(B,PA).
We denote the composite of this isomorphism with the functor {A,BI}*: K(A,P*B) °p and the attribute
{A,B]IA}*
split bifibration
E from A to
B to
A in
K°°, so that
from A to
P*, ~
B in
{A,BI }
by
.........~ SPL(A,B);
B when A = P*B is denoted by ~B" K becomes a s p l i t b i f i b r a t i o n
endow Ko°
with attributes.
A
E* from
Indeed, Ko°
is
also a cosmos. All previous precosmos theory dualizes. I t is consistent with our notation to write corresponding to
E(a,b): Y
E*(a,b): X
.>. P*Y for the arrow
~ PX in the situation of Proposition 1.
In particular,
for an admissible opspan X a A b y, we have hom*(a,b): X ... ~ P*Y defined by {x,YJhom*(a,b)}*
~
a/b.
164 An arrow j : A ~ B will be called laxfibre small when A and B are legitimate and there exists a strongly generating class G of legitimate objects such that, for all arrows g: G i B with G in
hOmB(1,g):B Theorem 28,
~ P*G and hom~(1,do): A If
j: A
G, j/g
, P*(j/g)
is legitimate and both
are coadmissible. j
, B is laxfibre small then any left extension along
of the form
J
A
~ B
PK
is pointwise. Proof.
Take
G as above.
We w i l l
applying Theorems 11 and 14 in
K° °
prove t h a t (c) o f Theorem 3 is s a t i s f i e d .
By
we o b t a i n an isomorphism
P*d 1
P*C ....
> P*(J/g)
P'B,
where ¥ g, ¥*d o are the l e f t a d j o i n t s P'g, P*d o, r e s p e c t i v e l y .
r
p*j
Now apply
(in
K(K,) °p
>
P*A
K; t h a t is r i g h t a d j o i n t s in and use the adjunction
Kc ° ) of
P* ~
P to
obtain an isomorphism
K(G,PK)
K()PK)
K(d~,l) ~ K(j/g ,PK)
K(j ,i)
~~K(A}PK)
where the vertical arrows are the right adjoints of
K(g,1), K(do,1). The diagram
obtained by replacing the arrows by their left adjoints (when defined) also commutes up to isomorphism. But the value of the left adjoint to the value of the left adjoint to the left extension of
K(dl,1)
fdo along dl. ~
at
K(j,I)
at
fd o is isomorphic to
f kg.
is
k.
So
So kg is
165 An object f: A
A is called small when YA: A
~ B is called kan when A
object of the span (f,A,y A) Corollary
29.
adjoint
f
Proof.
If
f: A
B to
PA exists and is an admissible opspan.
~ B is a kan arrow then
Pf: PB
~ PA has a left
~f.
PA is l e g i t i m a t e ,
exists by Theorem i i .
An arrow
is small, B is l e g i t i m a t e , and, the opcomma
from
and a right adjoint
Since
~ PA is l a x  f i b r e small.
bomB(f,1): B
~ PA is admissible;
so ~ f
By Proposition 26, the l e f t a d j o i n t of the functor
{B,PA]}: K(PA,PB) ~SPN(B,PA) exists at
(f,A,YA).
along YA exists.
By Theorem 25, the left extension
k: PA
By Theorem 28, this extension is pointwise.
~ PB of
hOmB(1,f)
So, by Theorem 27,
k = 3 f.# Left extensions in a precosmos have been discussed. cosmos we obtain results about r i g h t extensions. of arrows into objects of the form
P*K.
r i g h t extensions of arrows into
PK.
Suppose
j: A
The results about l e f t extensions
PK dualize to results about r i g h t extensions of
arrows into objects o f the form
Theorem 30.
From d u a l i t y present in a
More s u r p r i s i n g l y , we can prove results about
is an arrow with
~B
is coadmissible then each arrow
f: A .........~ PK
served by any arrow of the form
Pg: PK   ~ PK'.
B
legitimate.
If
has a right extension along (If
~P
w
hOmB(1,j) j
pre
is representable then
the converse of the last sentence holds.)
Proof.
By the dual o f Theorem 11,
existence of a l e f t a d j o i n t P*~
¥*j
hom~(1,j)
to
coadmissible is equivalent to the
P ' j : P*B .............> P*A. From the 2adjunction
P we have a commutative square
K(j ,I) K(B,PK) . . . . . . .
K(K,P*B)°P
~ K(A,P,K)
,,
~ K (K,p*A)°P
K(1,P*j) °p
166
But
K ( I , P * j ) °p 4 K ( I , ~ * j ) °p, so
f : A~PK
K(j,I)
has a r i g h t extension along
the n a t u r a l i t y of the above square in Corollary 31. hOm~A(1,Pf)
Suppose
j. K.
Then
So each arrow
The preservation property follows from We leave the converse to the reader.#
is an arrow such that
f: A ~ B
is coadnissible.
has a r i g h t a d j o i n t .
Pf: PA + PB
is legitimate and
PA
has a left adjoint
3f: PA+PB.
Proof. along by
Apply the theorem to the arrow Pf
Pf
gives an arrow
we have
3f ~
3f:
The r i g h t extension of
I : PB  ~ P B
PA~PB, and since this r i g h t extension is preserved
Pf.# f : A~B
Corollary 32.
Suppose
Pf: PB ~PA
has a left adjoint
§6.
Pf.
is an arrow in a uniform cosmos. 3f
and a right adjoint
Then
~f.#
Universal constructions. Whilst 2CAT with
PA =
[A °p, Cat]
is not a cosmos in our sense, we do have
pointwise l e f t extensions in the sense of Dubuc ( [ 6 ] with tion arrows.
V = CAT) and representa
So we can carry over the d e f i n i t i o n of " c o l i m i t " given in ~3 to 2
categories themselves.
Since ~
extension of an arrow into
is a strong generator f o r
2CAT, any pointwise l e f t
K along a representation arrow should be constructible
from the tensor product of arrows Fe7 :~~[A°P,coat] we i d e n t i f y 2functors out of
~
with arrows F:
A~K.
If
with objects of the target 2category, the
d e f i n i t i o n of tensor product (see ~4) amounts to the f o l l o w i n g . Given 2functors
o:
A°p +Ca~t, r: A   ~ K , the object
@~ ?
of
K is
defined up to isomorphism by an isomorphism K(e 0 r ,K) ~ natural in K.
[A°P,cowt](e,Er,K])
Note t h a t we could also ask f o r 2  n a t u r a l i t y in
K in which case we
167 say
O ©F
is 2enriched; 1)
Examples.
If
@© F = lim F •
A
t h i s i s automatic when
i s a category and
1 ~
Take
A
object
Let
9: A u\v
3)
fall
F: A
> K.
~
then
So pushouts
as s p e c i a l cases.
0: A°p
> £a~t denote the opspan ~ 6o ~  ~  ~i  ~ in Ccut. A u v i s a span A ~ S > B in K and @ ® ? i s the opcomma
>K
o f t h i s span.
Let
one object
A denote the simplicial category.
*
monad (A,s) @: A°p
@ i s the constant f u n c t o r at
t o be the t h r e e o b j e c t category w i t h two n o n  i d e n t i t y arrows thus:
2~I '
functor
is representable.
j u s t the usual c o l i m i t o f the f u n c t o r
c o e q u a l i z e r s , coproducts a l l 2)
K
and A(*,*) = A.
A 2functor
in the 2category
> Co~t is a monad in
K Oxt.
Take A to be the 2category with
F: A
; K can be i d e n t i f i e d with a
(using the language of [21]).
So a 2functor
We take this monad @ to be the monad called
AA by Lawvere in [15] pp1501. Then [A°P,cxt](@,[F,K]) So @®F in
K°P) 4)
~
[A°P,ccut](@,(K(A,K),K(s,1)))
is the construction
As
of k l e i s l i algebras
for the monad (A,s)
in
K (again see [21]).
Take
~
K(A,K) K ( s ' l )
~
(= construction of algebras
A
to be the 2category w i t h two o b j e c t s and n o n  i d e n t i t y arrows and f 0~1. A 2  f u n c t o r £: A ~ K is just a 2cell A~o~'~B
2  c e l l s thus:
v
in
K.
K(As,K).
We leave i t
to the reader t o f i n d the 2  f u n c t o r
(that is, a natural transformation)
such t h a t
g
9: A° p    * C a t
@ ® F = B[o  I ]
w i t h the f o l l o w i n g
universal property:  t h e r e i s an arrow and, i f
B
u n i q u e l y as We c a l l 5)
~K B
B
~ B[a  I ]
such t h a t the 2  c e l l
i s an arrow such t h a t > B[a  z ]
B[a  1 ]
ka
p@ i s an isomorphism,
i s an isomorphism then
k
factors
~ K.
the l o c a l i z a t i o n
of
B at
~,
There i s a dual n o t i o n o f cotensor which g e n e r a l i z e s " l i m i t " .
As examples
we o b t a i n p u l l b a c k s , e q u a l i z e r s , p r o d u c t s , comma o b j e c t s , e i l e n b e r g  m o o r e c o n s t r u c t i o n s and o p l o c a l i z a t i o n s w i t h i n a 2category.~
168
Theorem 33.
A representaSle 2category with a 2terminal object admits the follow
ing constructions: (a) finite limits; (b) the construction of algebras; (c) oplocalization. These constructions are all 2enriched. Proof.
I t is well known that pullbacks and a terminal object imply a l l f i n i t e
limit~.
Part (b) is intimated by Gray ~ 0 ] and a proof w i l l appear in his f o r t h 
coming book [11];
we also discussed i t in [12]. f A~ B .
For part (c), take a 2cell structure on
isoB
......... ,
hom B
homB such that, for a l l
K, the f u l l image of the composite
K(K,iSOB)
........ K(K,hOmB)
is precisely the f u l l subcategory of which are isomorphisms.
One r e a d i l y checks that oe
Using l i m i t s and the category object
g (= ~B in [22] Proposition 2), one r e a d i l y constructs an arrow
~
K(K,B) z
K(K,B) ~ consisting of the arrows
K
> B
Then form the pullback
e
i
~ ii°B
A ......
~
homB
is the universal arrow into
A with the property that
is an isomorphism.# Familiar techniques prove the following.
T~orem 34.
Suppose @: A°p
exists. If ~: K
~ Cat,
F:
A
~ K are 2fun~tors such that 9 ® F
> 11 is a 2functor with a right 2adjoint then o@(~r)
~
~(O@F).t/
169 As promised in the i n t r o d u c t i o n of [21] we shall prove the r e s u l t which relates the eilenbergmoore construction to the i n t e r n a l sheaves of a c e r t a i n type on the k l e i s l i construction.
Recall that
eilenbergmoore object
E (= As )
([12]§3.3), f o r a monad (A,s)
in
K, the
is defined by the condition that there is a diagram 1
u
E
flu
A
• A
s such t h a t
(u,y)
is the universal s  a l g e b r a .
Dually, the k l e i s l i o b j e c t
K (= As)
is defined by the c o n d i t i o n t h a t t h e r e is a diagram A
s
~A
K
J,K 1
such t h a t
(j,,))
is the universal s  o p a l g e b r a .
K is the coeilenbergmoore object in Theorem 34 y i e l d s that (PA,Ps)
in
K;
K°°°p.
Note t h a t So i f
(A,s)
is a comonad and
P has a l e f t 2  a d j o i n t ,
PK is the coeilenbergmoore object for the comonad
that is: PK ,,
1
)PK
PA
>PA Ps
is the universal Theorem 35. that
(A,s)
P: Kc°°p
Pscoalgebra.
Suppose >K
K
is a representable 2category endowed with attributes such
has a left 2adjoint.
Suppose
is a monad for which the kleisli object
the eilenbergmoore object for
(A,s)
K
A
is legitimate and that
exists in
Ko
An object
if and only if there is a pullback
E
is
170 E
, PK
Pj
A i°PO0~,
For any arrow
a: X    ~ A
YA
~PA
we have b i j e c t i o n s A/a
~X
X
el
Yoneda a
X
~,,
PA
Ps
One r e a d i l y sees t h a t coalgebra; (Pj,/~) x: X
(a,~)
is an s  a l g e b r a i f and only i f
(YAa,O)
is a Ps
the diagrams j u s t t r a n s l a t e n a t u r a l l y through the b i j e c t i o n s .
But
is the universal Pscoalgebra, so in t h i s case t h e r e e x i s t s a unique arrow > PK such t h a t
g j . x = yA a
and
~.x
= O.
In o t h e r words, we have a natural
bijection
X
a
~A
A
X
x
A
~PK
~PA YA
between such salgebras and such commutative squares. U The above theorem represents o n l y one amongst many ways in which the various "limits"
are r e l a t e d .
One would l i k e to o b t a i n the k l e i s l i
eilenbergmoore since the l a t t e r
does not r e q u i r e c o l i m i t s .
c o n s t r u c t i o n from the There are two approache~
171 The f i r s t
is to take the l e f t a d j o i n t
A J >K k
E where
k
The monad (A,s)
eilenbergmoore object t e l l us that object f o r
j
u
and factor i t
is " b i j e c t i v e on objects";
but
The second is a general one which applies to any
gives a comonad (PA,Ps)
and we can form the
X f o r this comonad. Now we need a "recognition theorem" to
X is equivalent to (A,s).
~ E to
is f u l l y f a i t h f u l and
when do such f a c t o r i z a t i o n s exist? "colimit".
f: A
PK f o r some K.
Then K should be the k l e i s l i
The f o l l o w i n g "recognition theorem" does not seem good enough.
One would hope to be able to generalize the work of Mikkelsen on complete atomic boolean algebras in a topos to improve the r e s u l t .
For enriched categories the next
theorem appears in [ 5] p189, Theorem 36.
In a precosmos suppose
K is legitimate and suppose Z: K  ~ lexz
an admissible fully faithful~ dmnse arrow such that by
homx(Z,1).
Proof.
Then
X
is equivalent to
X is
exists and is preserved
PK.
We have the f o l l o w i n g l e f t extensions.
K
z
X
K
YK
~PK
/
YK
By Proposition 17, sion. z
But
is 1.
z
f ~
lexz
lexz 4 hom(z,1) and so lexz preserves the f i r s t l e f t exten
lexz.y K~ z So
~"~/hom
and
z
is dense so the l e f t extension of
lexz.y K along
lexz.hom(z,1) ~ 1. A s i m i l a r argument proves hom(z,1).lexz ~ i.~
172 ~7. I)
Examples
Ordered objects i n a topos A span (u,R,v)
given arrows f o r the f u l l
from
A
to
w,x:C ~ R, i f
B
in a category
uw = ux
subcategory o f
and
A
vw = vx
an ordered s e t .
An ordered object in
span (do,A1,d I)
is a r e l a t i o n ;
for
A
then
w = x.
A
A
to
t h a t an arrow f: A ~B
is a category o b j e c t
A
(d1,Az,do).
fo = f .
Since t h i s
f
Ao.
(I,A,1).
Given ordered o b j e c t s
is uniquely determined by
is
f o r which the Objects Write
A,B,
f
A
A°p
we say
is order preserving when there e x i s t s a f u n c t o r i a l
f:Ao ~ Bo
with
There i s a t most
the span i s then c a l l e d an order on
w i t h the reverse o r d e r
RcZ(A,B)
B; t h a t i s , RcZ(A,B)
are regarded as ordered o b j e c t s v i a the d i s c r e t e o r d e r
Ao
Write
SP~V(A,B) c o n s i s t i n g o f the r e l a t i o n s .
one arrow o f spans between any two r e l a t i o n s from
of
is c a l l e d a r e l a t i o n when,
arrow
we o f t e n w r i t e
f : A ~ B. The f o l l o w i n g d e f i n i t i o n
is e q u i v a l e n t to t h a t of LawvereTierney [ 1 7 ] .
An
(elementary) topos is a category
E which has f i n i t e limits and, for each object
has an o b j e c t
EA from A to
PA and a r e l a t i o n
A,
PA satisfying the following
"powerobject" condition: given a such t h a t
relation
R from
A
to
B, there e x i s t s a unique arrow
R ~ h*~EA.
Given an arrow
f: A~B,
define
In t h i s way we o b t a i n a f u n c t o r
f: PB ~?A P: E°p~ E.
P A x PA.
by the condition
The composite span
PA also comes equipped with a p r o j e c t i o n i n t o to
h: B ~ P A
A
~B°f = (Pf)*°EA. * EAOEA
from
PA
and so leads to a r e l a t i o n from
A
This r e l a t i o n corresponds under the p o w e r  o b j e c t c o n d i t i o n t o an arrow
^: P A x p A   + p A .
The e q u a l i z e r (d° 1
dl
CA ~
defines an order
(do,CA,dl)
^ ; P A x PA ........... ~ PA proj l 
on
PA.
Henceforth we s h a l l w r i t e
PA
for this order
ed object. Let
to
K denote the f u l l sub2category of
CAT(E) consisting of the ordered
173
objects.
Each of the categories
K(A,B) is an ordered set.
The essential
property of the order on PA is that r B
PA i f and only i f
A\
R~ S
s
where
r,s
correspond to the r e l a t i o n s
condition.
Note that
particular,
~A is
K AI
R,S
from
A to
B
under the powerobject
is a representable 2category with f i n i t e 21imits. with the order
In
(~1,AIoAI,~I).
There is a kind of l a x 2  1 i m i t which we did not mention in ~6 but which can be constructed in any representable 2category with f i n i t e 21imits.
Given any ordered
pair of arrows f , g : A~ ~B, t h e i r s~equalizer (Lambek [13]) is a universal diagram of the form
k E
~A
A
,B
,
f A construction
for the subequalizer
E
is t h e p u l l b a c k
5
~ #B
A
BxB
~
We now wish to extend our functor For
A
in
K, l e t
.
P: E°P~K
PA be the subequalizer of
to a 2functor
P: Kc°°P*K.
Pdl, Pdo:PA0 ~PAI.
inc PA
~ PA0
inc I
~
PAO
IPdo ~ PA1
.
Pd1 The 2functor structure o f Suppose A,B t i o n from
A
to
P is induced using the enriched " l i m i t " property.
are objects of
K.
An ideal from
B which is a r e l a t i o n .
of
SPW(A,B) consisting of the ideals.
Bo
in
E, there is a unique order
Let
A to
IdX(A,B)
B
is a s p l i t b i f i b r a 
denote the f u l l subcategory
Given a r e l a t i o n (u, R0,v)
( d o , R l , d l ) on
Ro
from
A0
such that (u,R,v) is a
to
174
relation from A to
Proposition#7.
B in
K.
So Id£(A,B)
is a subcategory of
ReZ(Ao ,Bo)
The oomposi~ f~ctor
K(B,PA) K ( I ' S n ~ K(B,PAo)c
~ E(B0,PAo)~ RcZ(Ao,Bo)
induces a~ equivalence of categories
K(B,PA) ~ ~dZ(A,~).
Proof. Each functor in the composite is clearly f u l l y faithful so i t remains to show that the composite is surjective up to isomorphism onto the ideals. Proposition 12 of the last paper, a relation A to
R from Ao to
By
Bo is an ideal from
B precisely when BIoR ~ R and R°AI ~ R (the extra conditions are diagrams
in the ordered set RPZ(Ao,Bo) and hence automatically commute). Let correspond to
R under
E(B0,PAo) ~ ReZ(Ao,Bo).
translate to the following conditons on ~Bo
> PAo
Bo
1r
~
B0
~A 1 ~ R
r
BL
[
BIoR ~ R,
r.
dI
do
The conditions
r:Bo~PAo
rl
@A0
,
PAo
r
I Pd0
~I
~ PAl Pd:
The f i r s t of these says precisely that second says precisely that
r: B +PAo
is order preserving and the
r: B+PA 0 factors uniquely through inc to y i e l d
r: B ~ PA.~ An arrow of spans between ideals from of s p l i t b i f i b r a t i o n s from an ordered set.
B is automatically a homomorphism
B since the homomorphism axioms are diagrams in
So Id~(A,B) is a f u l l subcategory of
is a subpseudofunctor of of
A to
A to
Id~, the inclusion
SPL(A,B).
SPL (see early @1). With t h i s pseudo f u n c t o r i a l i t y
Id~(A,B) ~R~(Ao,B0)
becomes pseudo natural in
composite functor of Proposition 37 is c l e a r l y pseudo natural in that the equivalence
Theorem 38.
K(B,PA) ~ ~cL~(A,B)
The 2categoz~
Furthermore, I ~
K
is pseudo natural in
of ordered objects in a topos
the attributes are precisely the ideals.
A,B.
A,B.
The
I t follows
A,B.
E
is a cosmos in which
175 Proof.
We have already seen that the f u l l y f a i t h f u l functor
is pseudo natural in ideal
EA from
A to
A,B.
Set B = PA and evaluate at the i d e n t i t y to obtain an
PA which consequently endows K with a t t r i b u t e s .
We have
immediately that the attributes are the ideals so the precosmos condition is s a t i s fied.
One r e a d i l y v e r i f i e s that
P has a l e f t 2adjoint
P*
given by
p.A = (pAoP)°P.# For an ordered object
A, the s p l i t b i f i b r a t i o n
r e l a t i o n and hence an ideal. Corollary39.
@A from
A to
A is a
So K is a uniform cosmos.
For each orderpreserving arrow
f : A +B, the orderpreserving arrow
Pf: PB *FA has both a left and a right a~oint.#
2)
The preSp~nier constuction as a
P.
The 2category Simp and 2functor
P presented here are taken from unpublished
work of DayKelly on categories l i k e categories of topological spaces. A function
f : X+Y
between sets
through the onepoint set i . functor
I I: A+Set
X,Y
A simple category is a category
SC2. the image of
I la, b
SC3. there is an object
is in
a, a' in A, we write
A(a,a').
A together with a
satisfying:
SC1. I la,b: A(a,b)   ~ S e t ( l a l , l b l )
For objects
is called constant when i t factors
A functor
contains a l l the constant functions;
a of a ~ a'
f : A+B
is an inclusion of sets;
A with when lal
lal ~ o. =
la'I
and
Iial:
la l~la'l
between simple categories is simple when the
following diagram commutes. f A
Let
~B
Simp denote the 2category whose objects are simple categories, whose arrows
are simple functors, and whose 2cells are natural transformations. A,B
in
Simp, the category Simp(A,B)
Then, f o r each
is an ordered set; indeed, there is a
176 f natural transformation
A~'+~ B
p r e c i s e l y when
fa ~ ga
for all
a
in
A.
g Pullback in Simp is t h a t of CAT and Simp is a representable 2category. simple functors
A~r D,
BS
D,
The objects are p a i r s (a,b) with arrows are pairs
the simple category a
in
A, b
(m,B): (a,b) ~ ( a ' , b ' )
in
where
r/s
B and
For
is defined as f o l l o w s . ra ~ sb
~: a   ~ a ' ,
in
D.
~:b   ~ b '
The are in
A ,B
r e s p e c t i v e l y , and the f o l l o w i n g square commutes. ra 
IIa
]+ sb
ra' IT~_, sb'
[al
Let
A
An o b j e c t
x
f o r each
a
be a simple category. of in
A is a set
FA
indeed a 2functor 4.3
(4.6)
forgo,
sending
di:
4. 3
(A,£)J(B,A)
~ A'SB
SB
~a'gB
gB
~B
to check that ~ respects
vertical
of 2cells in CATf*T and CATfT; thus J
of CATf*T x CAT~T
CAT#*T
(3.14),
onto its factors,
give new forgetful
and is
and
composed
2functors
x CATfT ~ CAT ,
(B,A)) to A and B respectively.
clear from (4.12) Proposition
~ AB k
2functors
forg1:
((A,£),
aB
AB
of the form (4.9).
The projections
with the forgetful
k
~,FA
It is very easy from (4.15) composition
~ (~,T)#(S,~)
is
~A
horizontal
transformation
The following is
(4.15):
The functors do:
(A,F)J(B,A)
~ B are the components
~ A and
of 2natural transformations
do: J ~ forg O and d1: # ~ forg I. D
5. 5.1
THE 2FUNCTOR
o AND THE EMBEDDING
Form the composite
CATf*T x
CAT~*CAT ~ [CAT,CAT]
2functor
T~~
CAT[*T
×
CATfT
i × r • where r m
(I,~B~),
: Y ~ CAT~T etc.
is the 2functor
Our interest
write the above 2functor (5.1)
o: CATf*CAT
~ CAT, #
(3.13)
is in the special
as
x CAT ~ CAT.
sending B E Y to case Y = CAT; we then
201
We usually
abbreviate
value on morphisms
its value
(A,F)oB on objects
or 2cells we write
except that we are sometimes
interested
to AoB ; for its
(T,T)oS and (q,8)o~ in full; in the restriction
CAT/CAT 0 × CAT, where T and 8 are identities,
of o to
and then we write simply
ToS and qo~. An object of AoB is a pair
(A,X) where A C A and X is a functor
FA ~ B; we write this object as A[X]. pair
A morphism A[X] ~ A'[X']
is a
(a,x) of the form
FA
A
(5.2)
a
ra
A' here a is a morphism the morphism
8;
FA'
S
in A and x is a natural
transformation.
B~
B', the functor
We write
(5.2) as
A[ X]
(5.3)
l
a[ xl
A'[X'] If (T,T): (T,~)oS:
AoB
~
(A,F) ~ (A',F') A'oB'
and S:
sends, as a special
case of (4.12),
the morphism
(5.2) to (5.4)
TA
Ta I TA'
F'Ta I
TA ~Ta
~~FA ~
~B'
F'TA'
(T,T)oS ~ (T,T)~S has,
component
X
!i"~~XX B
TA , (T,T) ~ (T,~) and ~: S ~ S, the natural
If (q,0): (q,O)o~:
F 'TA 
transformation
as a special case of (4.15),
the A[X]
202
(5.5)
TA
F'TA S
HA
r'r~ A
FA
~B~ B '
...... X
F 'TA
TA
We have
(5.6)
do:
sending ent
as
(5.2)
in
where
fOrgo:
5.2
The
gives
there
K ~
As in P r o p o s i t i o n
4.3
this
is the
compon
° ~ forg O ,
CAT[*CAT
in p a r t i c u l a r
functor
transformation
x CAT
3category
2functors
the p r o j e c t i o n
to a: A ~ A'.
do:
TA
AoB ~ A
of a 2  n a t u r a l
(5.7)
§4.1

~
sends
CAT
of 2  c a t e g o r i e s
is a b i j e c t i o n
[L,M]
Under
.
this
((A,r),B)
to A, etc.
is of c o u r s e
between bijection
cartesian
2functors the
closed;
K × L ~ M and
2functor
o of
(5.2)
a 2functor
(5.8)
CAT~*CAT ~ [ C A T , C A T ] ,
@:
to wit
(5.9) Our
¢(A,F) first
Theorem
main
5.1
result The
Proof.
Take
components
GB:
unique
= Ao,
= (T,T)o,
¢(n,8)
= (~,8)o.
is:
2functor
a 2natural AoB ~ A'oB.
} is 2  f u l l y  f a i t h f u !. transformation
G:
We are
that
to
show
(A,F)o
~
(A',F')o,
G = (T,T)o
with
for a
(T,T). The u n i q u e n e s s
follows
(5.1o) and
¢(T,T)
that
from
(5.4)
of
is i m m e d i a t e ;
that
Gri(A[irA]) GrA , sends
(T,T)
= TA[TA],
the m o r p h i s m
for
if G is
(T,~)o
it
203
(5.11)
A
FA
al
Fa
rA'"
A' of AorA'
FA'
to the morphism TA
(5.12)
TA
F 'TA
r'Ta
Tal TA' of A'°FA';
(5.13)
~FA
]I
v
F'TA'
Ta
[
ra
~
U i
rA'
~FA'
TA ,
that is to say,
GrA ,(a[lra]) = Ta[~a ].
So by (5.10) and (5.13) G uniquely Now let any 2natural that by the naturality
determines
G be given,
T and T.
let AtX]
of G we have commutativity
E AoB, and observe in
GFA
(5,14)
AorA
~ A' rA
A'oX
AoB
B.
If we define TA,T A by (5.10) and evaluate object A[IFA]
(5.15)
we get
a~(AEx]) = TAEX.TA].
In particular, this said,
GFA , (At Fal)  TAIFa.'rA],
the domain of (5.12).
it now makes sense to define Ta,T a
Next, replace A,X by A',X' the morphism a[ira] (5.16)
both legs of (5.14) at the
of (5.11).
With
by (5.13).
in (5.14) and evaluate both legs at
By (5.13) this gives
GB(a[ Ix, .ra ] )  Ta[X' .T a] .
204
Now let
a[x] be the morphism (5.2) of AoB.
By the 2natural
ity of G we have commutativity in G AorA
(5.17)
AoX
rA
~A oFA
Aox
A'orA'
A'oX
AorA'
/
~
A'oX'
~ A ' oB
GB
Taking the A[IrA]component of each leg we get, using (5.10) and (5.5), (5.18)
GB(1A[ xl )  1TA[X.~ A] .
But the a[x] of (5.2) is the composite A[X]
~
A[X' .ral
1A[ x]
~ A ' [ X ' I
,
a[ i x , . r a ]
and G B is a functor; so GB(a[x]) is, by (5.16) and (5.18), the composite TA[X.T A]
~~ TAIX'.Fa.T A] ~ TA'IX'.TA,] ; ITA[X.T A] Ta[X'.T a]
that is, (5.19) GB(a[x])
is
TA
Ta
F'TA
TA
~FA
I I I r'Ta
TA'
F'TA'
~FA'
TA ,
Using the fact that G B is a functor, we now easily get that T is a functor and ~ a lax n.t.
by applying GFA to IAIllr A and by applying
205
GFA,, to the composite A[ra'.ra]
~A'[ra']~A"[IFA,,].
a[ i] Then by comparing
It remains let
(T,T),
(~,~):
modification
a'[ I]
(5.19) with (5.4) we see that G B is indeed (T,T)~8. to prove that ¢ is fully faithful
(A,F) ~ (A',r')
with components
YB:
that ¥ = (n,8)o for a unique
and let y:
component
(5.20)
(T,T)o ~ (T,Y)o
(T,T)oB ~ (T,~)oB.
So
be a
We are to show
(0,0).
If ¥ is indeed of this form, A[IFA]
on 2cells.
it follows
from (5.5) that the
of ¥FA is
(~rA)A[irAl:nA[e A],
which proves the uniqueness
of n and 9.
If now for any ¥ we define
and 9 A by (5.20), we have because y is a m o d i f i c a t i o n
commutativity
HA in
(T,T)oFA
AorA~YrA
A' rA
A ToX
Ao X
(T,~)oB Ao B
YB
oB,
(T,~)oB and calculating
(5.21)
the A[iFA]component
(yB)ALX]
in agreement
with
: nA[XeA] , (5.5).
indeed a modification, YFA'
Proof
That n is indeed natural,
5.2
CAT/T
and that 0 is
follow at once when we express
for the particular m o r p h i s m
Corollary
of both legs gives
and
By the remarks
CAT/*T
(5.11) of AFA' are indeed
the naturality
of
D
2catesories.
in §3.1 and §4.2, with the above theorem.
U
206
5.3
We d e v o t e
this
full
embedding
First, note
that
I is the
F = rn~
for
some
discrete
image
the
of ~.
with
were
in the
that
this
[n,]
sending
part
I have because,
while
h o w to fix image
clear
that
[n,]
for
D is not
both
DB
form
D need
not
that
form.
[n,] .
D3 = 38.)
be in the
explicitly,
for
DI is at any rate to [n,B]
I think
image
Now
of p a t h 
D2 = 24 and
equivalent
in the
n.
2 is the
of the
can be g i v e n
of this
of s o m e t h i n g
where
not
it is c l e a r
to m a k e
some
m is the n u m b e r
with
which
There
is e n o u g h
D is a q u o t i e n t n is ~
where
of a d o c t r i n e
the d o c t r i n e ,
of ~; but
be
is c e r t a i n l y
no m is c o n s i s t e n t
I; this
A ~ I, so that
[[B,2],B],
B to
B is d i s c r e t e ,
2functor
image
where
see this,
CAT ~ CAT we have:
D:
then
it is also
DI ~ I m u s t
2 objects,
when
Consider
to
To
~ [n,B];
a categorywithcoequalizers. equivalent
on
A a DI.
But
of ~ w i t h
of n; and
Even
CAT].
of [ C A T
if D ~ (A,F)o
n.
CAT ~ CAT
= [m,B]
components
then
category
category
[n,Bl
the w h o l e
informal,
2categories.
So for a 2  f u n c t o r
~ I and
image
D:
~ [CAT,CAT]of
is not
category.
(l,rn~)oB
2functor
comments,largely
~ A,
if DI
so any D in the
to some
clearly
unit
that
(5.24)
(For
image
If D ~ (A,F)o
follows
the
section
@: C A T f * C A T
(A,F)oI
(5.23) It
its
we have
(5.22) where
this
if D
however
of @, to wit
of
.
no real
idea
I can p i c k
F in terms
how big
out
the
image
the p u t a t i v e
of D, and
hence
have
of ~ in fact
A by
(5.23),
no test
is;
chiefly
I don't
see
for D to be in the
of ~. We are
composition products
and
going
to show
of 2  f u n c t o r s . coproducts.
in What
§7 that else
the
image
is it c l o s e d
of @ is c l o s e d under?
under
Certainly
2O7
It suffices
to i l l u s t r a t e
CAT~*CAT of (A,F) and and
A',A)oB ~
(A ×
(A,F)
and
(A',F')
(A + A',a)oB ~
But
(A,F)oB is
it would
AoB
as
The product A(A,A')
coproduct
in
= FA + F'A';
CATf*CAT
in
AIA = F and AIA'
has
of
= F'; and
my r o u g h
constructed
a 2natural
AoB.
pointwise,
(A,F)~(I, r B ~ ) ,
B~, where
adjoint,
G: D ~ ~(A,F)
such a G B would
that
that
seems
false.)
preserved.
modulo
size
: (A,F) o, with
4.1 and the d e f i n i t i o n
correspond
B~ is the
which
they are p r o b a b l y
a left
By P r o p o s i t i o n
indicate
them it w o u l d mean
on the other hand,
Consider
chiefly
to our m a i n purpose.
calculations
(If @ p r e s e r v e d
were
equalizers:
not r e l e v a n t
seem to have
FP ~
The
CATf*CAT
@ would
GB: DB ~
er with ~:
where
whether
them.
If it has coequalizers,
components
where
x (A',F')oB.
equalizers,
of lax n . t . ' s
considerations.
(A × A',A)
be a long b u s i n e s s
wou l d not preserve
For then
ones.
+ (A',F')oB.
checked
if it does have
equalizers
is
(A + A',A)
(A,F)oB
I haven't because
(A',F')
by b i n a r y
of
to P: DB ~ A togeth
composite
DB ~ I ~ CAT,
rB~ that
is, the
bijection selves in
constant
with m o r p h i s m s
would
CAT~*CAT
(5.25)
of
in
is, by
seen to be 5.4
in
other hand
in the
CATf*CAT
already
(A,F), (5.24)
(B,A)on. We shall
CAT,
take
not n e c e s s a r i l y
The
G's them
~B(DB,B~) ~
(A,F)
that
lies
~ is as far as it can be from
that
any D E [CAT,CAT]
in the
latter.
For
with
a
if
~ CATf*CAT((A,F),(B,a))
and Yoneda,
always
CAT~*CAT.
in
exists.
sense
(A,F)
So D ~
in
the G B are
w i t h the m o r p h i s m s
coend
adjoint;
In other words
~ (A,F)
it seems
[CAT,CAT]((A,r)o,D)
2naturally (5.25)
(DB,B!)
if the i n d i c a t e d
a right
2reflexion
at B.
then be in b i j e c t i o n
On the having
functor
: (I, rr~). Dn;
while
Then the the right
left side
side of is easily
(B,A)o.
henceforth
use T to denote
full on 1cells,
but always
a sub2category taken
to be full
208
on 2cells. We have an obvious
(5.26) which
inclusion
CATf*T ~ CATf*CAT composed
(5.27) It is clear precisely
with
¢ gives
¢T:
CATf*T
~
that
(5.26)
is full on 2cells,
when T ~ C A T
6.
6.1 closed
in [ C A T , C A T ]
Ao(Bo)
~
CAT
~
is;
so that
goal
under
CAT).
(AoB)o;
[CAT,CAT].
here
composition
We shall AoB
of A and
B. The c o n s i d e r a t i o n s
n = £A of A E A is a kind
~
of objects
CAT
of B.
If these A: B ~ CAT,
is the composite
suggest
of A.
that
CAT[*CAT is
is, of
that
so denoted
in 55, but now
from the a u g m e n t a t i o n s
that
An object
objects
the a u g m e n t a t i o n A[X]
of B also have arities,
then A[X]
AX: n ~ B ~ C A T ,
should
of A o B
and where
n(m)
generalizes
In fact
ization
construction
is p r e c i s e l y
the G r o t h e n d i e c k
n and a functor
m: n ~ C A T
over n, which we may
call n(m).
6.2
define
We therefore
8:
the
the G r o t h e n d i e c k
CAT,CAT ~ CAT
the a p p r o p r i a t e assigning
corresponding
2functor
of
provided
have an arity n(m),
n(ml,... , m n) = m I +...+ m n of §1.2.
(6.1)
(5.27).
of A and X: £A = n ~ B, so that X is a kind of nad
augmentation
category
of
(that
showing
derived
of 51.2 above of "arity"
is true
in §7 b e l o w
do this by
AoB
and is full on 1cells
CONSTRUCTION
is the category
with an a u g m e n t a t i o n
objects
same
is the proof
provided
consists
the
THE G R O T H E N D I E C K
Our next main
2functors
,
fibred
by an
where m the generalto a category
209
by setting (6.2)
@ = rill,
where o f c o u r s e /
is
the 2functor
We a l s o u s e , where c o n f u s i o n
of
(4.9)
is unlikely,
with
T put equal to
the "parenthesis"
CAT.
notation
suggested above: namely (6.3)
@(A,£)
= A(£),
An object (that domain
@(T,T)
of A(F)
= T(T),
We write
= q(e).
is a pair A,Y where A E A and Y: I ~ FA
is, Y E FA; but it is useful ~).
8(q,8)
this object
to write
as A(Y).
Y as a funetor A morphism
A(Y)
with ~ A'(Y')
is
a pair a,y of the form (6.4)
A
aI
which
I
Fa
is a fancy way of saying that y is a morphism
We write this morphism If (T,T): functor T(T)
sends
(6.5)
TA
ra.Y ~ Y' in rA'.
as a(y) .
(A,£) ~ (A',r')
in CAT/CAT,
then by (4.12)
the
(6.4) to TA ?r'TA
Ta
~a TA'
r'Ta ~F'TA'.
TA , If (q,e):
(T,T) ~ (T,T)
has by (4,15)
(6.6)
its A(Y)
in CATfCAT, component
the natural
given by
~
TA I
~A
transformation
y
~
F'TA
rA
F'~A
q(8)
210 We have as in §4.1 the p r o j e c t i o n functor
(6.7) sending
d1: A(r) ~ A (6.4) to a: A ~ A'.
As in P r o p o s i t i o n 4.3 this is the
component of a 2natural t r a n s f o r m a t i o n
(6.8)
d1: @ ~ forg,
where forg: A, etc.
CATfCAT ~ CAT is the forgetful
2functor sending
(A,F) to
W r i t i n g ~ for the arrow category 0 ~ i, we can regard 9,
forg, and d I in (6.8) as c o n s t i t u t i n g a 2functor @: CATfCAT ~
(6.9) sending
[~,CAT]
(A,F) to dl: A(F) ~ A, etc.
We can call O the a u g m g n t e d
G r o t h e n d i e c k 2functor. 6.~
The results we need about the G r o t h e n d i e c k c o n s t r u c t i o n are
most easily derived from the following theorem, which is a slight extension of Gray's "YonedaLike Lemma" on page 290 of [5]. T h e o r e m 6.1 Proof.
CAT,CAT ~ [~,CAT]
The 2functor 9:
is 2fullyfaithful.
Let
A(r)~
(6.10)
~ ~
A
A'(r,)
~
A'
T be
a morphism
If
in
M is
[~,CAT].
indeed
T(~)
We h a v e
we
to
show
conclude
(6.11)
M(A(Y) ) = TA(TAY) ,
(6.12)
M(IA(Y) ) = ITA (TAy) ,
which fixes T A on objects and morphisms.
that
from
M = T(T)
(6.5)
with
for
a
a
unique
= 1 A that
M o r e o v e r if we define T A by
211
(6.11) and composite
(6.12), the fact that M is a functor and the fact that the of iA
AtX'
¢1''" X'¢n] aiX,1...X,n,~ 1 A'EX' l ' ' "
X~].
The reader will have no trouble expressing (9.24) and (9.26) in this expanded notation. Since the special case CAY/~ has been treated in detail in [5] and [6], and since there is not much to change, I shall give only an outline in this section.
There are some mild notational differences
from [5] and [6]; there ¢ was always an isomorphism and (10.2) was replaced by xi: X
~
X'i, which is impossible here.
That, however,
had the effect of making (10.6) conformable with the convention that X' stands for IX,; here we have to sacrifice something and we choose it to be this.
244
10.2
What makes
ness
of the
say b e l o w most
images
would
practical Let
CAT/~ ".
the
FA u n d e r
apply case
case
the
equally
that
the a b o v e
well
with
T[S1,...,
S n] ~ T ( S 1 , . . . ,
S n)
for
A club club; are
it m a y
clubs
of the
CAT/~
above
starting
is that,
that
Let
construct
are
forms
free
automatically inductively
the
objects
not
the basis
free
club
discreteof what
stick
we
to the
"club
"multiplication"
same
identity
(Q,F)
a discrete
(Of course further
consequence
the
objects
is false
that
we do in this
be given.
We
for
before
to
 which
construct
and
section.
It is easy
it g e n e r a t e s
and
CAT/cato,
that
it g e n e r a t e s .
there
K from generators
settle
~; and
CAT/~.
in
is c a l l e d
The
in
if K is a club,
same
SET/~.
in
and
SET/~
club
the
a club
of all
of
means
and the
discrete).
It is this
discrete
we
that
as a c a t e g o r y
IKI first
(P,F)
much
IKI ~ K is a club m a p
in c o n s t r u c t i n g
any o b j e c t
the
it f o l l o w s
as a o  m o n o i d
on the m o r p h i s m s .
10.3
is also
the
set
Q
by
(10.8)
& • Q; if P E p w i t h FP
(10.9)
P(QI''" We e m b e d
agree
that
over
S),
until and not
for any
if Q I ' ' ' ' '
Qn E Q t h e n
qn ) E Q.
FP(Q 1 ... qn)
augmentations
extension
: n and
P in Q by i d e n t i f y i n g
F~ = i,
that
is d i s c r e t e
that
but
"club"
of c o u r s e
inclusion
we can c o n s t r u c t
it is this
by
the
be r e g a r d e d
in
relations,
which
section
discreteness IKI,
that
is the
in fact
CAT~set,
to
for this
set of o b j e c t s
also
simple
operations.
so is its
it f o l l o w s
specially
augmentation;
of f i n i t a r y
us a g r e e
From
present
further
notice
a general
L and
to a club m a p
P(~
... ~),
= n ( F Q 1 , ... , FQn).
are d e n o t e d club
P with
one
"club
map"
To
Let
by F if no o t h e r
name
any m o r p h i s m ~ ~
L.
P ~
save
means
in CATf*~.
and e x t e n d
repetition,
one
in
us also
there
let
CAT/~
(i.e.
agree
that
is given.
L over ~,
F to Q
It
us
all
is c l e a r
is a u n i q u e
245
Now and
let
consider
clear
that
these
congruence and
discrete
then
A model : P ~
is of c o u r s e
always
ITI =
ISI
for the
A; here
~(~,I)
identity 10.4
~
or
objects
into
obtained
by
operation We
can
implies
implies
club m a p s
Q ~S
first
call
S
rT = rS,
MT = MS.
enlarging
the
It is
~ L,where
T ( S I , . . . , S n)
where over
A together
n = FP. ~;
This
the
S is
p to a
of s u b s t i t u t 
discrete
with,
club
for e a c h
is c l e a r l y
the
endointernalhom
a club;
and
{ A , A }, or a g a i n
with
P ~ P,
same
thing
( A , A}
so a m o d e l
of P is the
same
as an a c t i o n
Q oA ~ A;
so it
Qcategory.
of P,O
is such
a model
TpS.
A of P s a t i s f y i n g
It is t h e r e f o r e
an S  c a t e g o r y
S. example
~ and
is that
I with
p identifying
of a strict
F@ = 2 and
~(~,~)
$; the c o r r e s p o n d i n g
with
discrete
monoidal
FI = 0, and
~(~,
club
®),
category
Q
is s u b j e c 
~(I,~)
with
S is ~ itself,
~, and
with
augmentation. a discrete
and a r r o w s
and let
us be g i v e n as w e l l
graphmorphisms a clubmap.
 not
that
now replace L
I assert any
call
over that
such M
the
S, w h e t h e r
a graph of [5]
a morphism
D ~
so that
L; we can
club
let us be g i v e n
objects
M ~
Q~
obvious
or d i r e c t l y ,
M,
ToS
canonically
The most
Given
We m i g h t
A,
club
with
that
is a c a t e g o r y
discrete
ted to r e l a t i o n s
L such
ToS
p.
: An ~ A whenever
P has
that
to the
{ A , A}
a Qal~ebra A model
club
P
as a c l u b  m a p
is just
such
S = Q/~.
An ~
as a m o r p h i s m
thing
of
relations of
on Q
uniquely
respect
and
IPI
: Q ~
club
setting
P
a functor
M
factorize
~ with
generators
a relation
clubmaps
the q u o t i e n t
ion,
0 be
D (in the u s u a l and
of g r a p h s
S by
IDI.
~ such there
by g e n e r a t o r s
that
[6]) w i t h r
factors
club
~enerated
of g r a p h
objectset
for a club
composite
is a u n i v e r s a l
D ~ L
sense
: D ~ ~ extending
Consider the
and r e l a t i o n s
through
IDI = S, r
: S ~ ~.
L those
IDI ~ D ~
L is
such map
D ~ M into
a unique
clubmap
by S and
D.

In p a r t i c u l a r
a
246
a model f
of
(S,D)
means
an S  c a t e g o r y
: T ~ S in D, a n a t u r a l
in the the
sense
club
of
(9.20).
objects;
formal
first
~ = Ff,
formal
f($
... ~),
but
: ITI ~
A is just
by e n l a r g i n g
an i n s t a n c e
we also
i.e.
so the
extend
IsI of type
rf,
an M  c a t e g o r y
for
D twice,
of an f
bigger
P E
graph
to w h i c h
but
a partial
without
changing
: T ~ S in D to be a
...
O",
instance
g
...
We
pass
that
any m o r p h i s m
IDI ~
L is a c l u b  m a p ,
diagrams
in D'''
g
... R%n)
so d e n o t e d .
etc. are
of c o u r s e
We i d e n t i f y
graph
they
...
g itself
the
f with
D' e x t e n d i n g
D, to
r.
The
of
o monoid
O ~
L over
extends
...
T
~)
with
~
P($
~(g),
category
~ into
generated
D'''
a club
is not
S
...
~)
a still
sent
There into
by the yet
on it;
L, w h o s e
to a f u n c t o r
on D'''
by
...
and n o w have
can be d e f i n e d
are n e c e s s a r i l y
an e x p a n s i o n
F.
category
uniquely
are t y p i f i e d
define
f, to be an e x p r e s s i o n
extend free
clubstructure
which
: T ~ S in D',
of some
we a g a i n
extend
structure
~L;
IDI
: P($
to D''',
we a g a i n
the p a r t i a l
l)
identify
to w h i c h
respects
T(R~I
~ S ( R I ... Rn,)
F.
clear
by D,,,
the o b j e c t s
for an i n s t a n c e
IOl.
We t h e n D",
... R~n)
instances form a b i g g e r
an e x p a n d e d P(1
some
T(R~I
are the o b j e c t s of
and
(i0.ii)
for
we d e f i n e
and w h e r e
We next, of g,
a model
M we b e g i n
f(R I ... Rn,):
not
which
So such
Ifl
for e a c h
expression
(i0.I0) where
transformation
with,
M. To c o n s t r u c t
its
A together
O'''
graph
a club,
and
it is
restriction ~ L
are two
commutative
which
sorts
of
diagrams
247
T(l...h...i...i) (lO.12)
T(SI...S~...Sj...S n)
T(SI...Si...Sj...S n)
T(1...1...g...l)
T(l...l...g...l 1
T(SI...S~...S~...S n)
T(Si...Si...S~...S n) T(l...h...l...1)
and by
(10.13)
T(S¢I . . . . . . . . .
S$n )
f(Si...Si...Sn,) .............
T'(SI...Si...Sn,)
T(1...k...k...l)
~T'(sl...s~...sn,).
T(S¢~ ......... Sen) f(S1"''S!'''Sn')m Here (10.13) needs some explanation.
The left vertical arrow stands for
either leg of a diagram like (10.12); or the corresponding thing when there are more than two maps k in it; and these k's occur at all those indices j for which Sj = i.
The diagram (10.12) says in effect that
T is "functorial", and (10.13) that f is "natural".
It suffices in
fact to impose these diagrams not for all f,g,h,k E D''' but for f E D", g,h,k ~ D'; the more general cases then follow automatically. All this is rather more complicated than in the CAT/~ case of [61. If we write Funat for the class of diagrams of the form (10.12) and Nat for those Qf the form (10.13), and D'''/(Funct + Nat) for the quotient category with the same objects got by imposing these relations, it turns out that this last is the desired club M; for it indeed admits a
clubstructure in which, in analogy with (10.7), f(gl...gn ) is
defined to be the composite
248
(10.14)
T ( S I . . . S n)
~ T(S'~I...S'~n)
~
T ( g l . . . g n) 10.5
Now
suppose
diagrams
f,g
: T ~ S in M w i t h
a set
that,
o of such diagrams
morphisms
D ~
L over
be a c l u b  m a p ,
f(S'l...S'n,)
besides
the above,
Ff = Fg;
in D''';
~ into
a club
for which, in the
induced
say K, a q u o t i e n t  c a t e g o r y
of M , a n d
M by i m p o s i n g
expanded
instances
for the of the
in fine
call
P, w i t h r e l a t i o n s
generators
D, w i t h
A model
imposed
K the
A of
usual,
graph
IDI ~ D ~ L, the
L to
images
of
such
clubmap
relations is o b t a i n e d
that
K ~
all
any
L.
the
such
In fact expanded
as
consists
of all
the
a.
club
p, and
the r e l a t i o n s
o of
more
those
L or M ~
a club,
relations,
relations
generators
club
as e x t r a
+ Nat + Imp),
We m a y
the
itself
K = D'''/(Funct
satisfying
~
a set
~here is again a universal such L,
a; or b e t t e r ,
Imp, s t a n d i n g
only
requiring
N'''
of the r e l a t i o n s
(10.15) where
from
given
is p e r h a p s
consider
as R ~ M ~ K ~ L for a u n i q u e
K is o b t a i n e d instances
~lo.
we are
what
L~ still
T ~S
factorizes
or,
a n d we
f and g coincide for each f,g:
map
T'(S'I.~'n,)
~enerated
by the
function
the n a t u r a l  t r a n s f o r m a t i o n 
o.
(P,p,D,o) is of c o u r s e a m o d e l of (P,@,D)
the r e l a t i o n s
o; it
is the
same
thing
as a K  c a t e g o r y
A for
K. A typical
example
is that
of
symmetric
monoidal
categories~ where
P consists of ~ and I with F@ = 2 and FI = O; where p is vacuous; where D consists of a
:
e(~,~)
~
~(~,e),
a
: ~(~,~)
r
:
@(i,I)
~
i,
~
: i
c
:
® ~ @,
with
identity axioms, a,~
Fa,
permutation together
as i n v e r s e s
with
F~, of
Fr,
~
~
®(i,I),
F~ i d e n t i t i e s
2; and w h e r e
~(~,~), and
and w i t h
~ consists
Fc the n o n 
of the u s u a l
coherence
aa = i, aa = i, r> = i, ~r = i, e s t a b l i s h i n g
of a and r.
Here
the
club
K is in fact
in CAT/S;
249
other e x a m p l e s 10.6
leading
to clubs
It should now be clear that
"coherence explicit already
problem"
determination
of the club
a wordproblem,
: T ~ S in D'''
that
is, of k n o w i n g
of course diagram,
in terms
f and g.
: ® ~ @
with
to the
consists
To find the objects
easy,
and trivial
in the of K is
if P is vacuous.
that
is usually
much harder.
and it is a q u e s t i o n
of k n o w i n g
when
coincide "which
in K because
diagrams
components
in the
of the relations
commute".
is the c o n d i t i o n
of generic
For example,
in §1.4 above.
solution
(P,p,D,q)
is a w o r d  p r o b l e m
Ff = rg; this
suggested
a complete
K.
but usually
We know the generators, f,g
are
for such a structure
To find the m o r p h i s m s
c,l
CAT/~
in
A necessary
in (10.15); condition
for the w r i t a b i l i t y of the "natural
of a
transformations"
symmetricmonoidalcategory
Fc ~ rl; and the generic
is
case,
we have
components
(10.16) C ~ B A®B
~ do not where
form a closed
A
In
closed diagram,
the
in the club
diagr a m s
commute,
10.7 (P,p,D,~)
The
situation
K.
If one wants
one has only
considerations
calculation
to conclude
case
is that
F is faithful.
: A ® A ~ A ® A form a
in a t y p i c a l model,
of §§10.3
described
even when
(P',p',D',~'). for example
KoA on A,
when we know K.  10.5
there
that
and makes
such s p e c i a l i z e d
to look at the free model
of the word,
relations
c,l
to know w h i c h
we also know c o m p l e t e l y
sense
commute"
for f = g, i.e. where
specia ! components
may be r e c o g n i z e d
generato~and
"all diagrams
case.
~ L of the kind
in our current
The
but this does not commute
no sense
which of course
B
sufficient
a rare
(10.16)
®
diagram.
Ff = Fg is also
It is of course
@ A
~
leads
assert
to a club map
is to say, L itself
that a "map"
in
CAT/~.
K ~ L This
is given only by
Thus we are able without
the existence
of a club map
K ~ L
250 over
~ when
symmetric
monoidal
monoidal
clubmaps may
also
when
together
very
observe
in
is in c o n n e c t i o n
by
the m o r p h i s m s itself
So an o b j e c t of the
we
that
as
observation
in
law
monoidal
(i.i)
in §9.5 lie
to leave
for
symmetric
symmetric
law,
we h a v e
i that
above. that
not
CAT/~.
in
(lax)
of the
considerations
(~,~)
doctrine
in the
2category
which
are
lax n . t . ' s
but
should
get
sense
K o for a club
FUN the not
of
: A ~ A' b e t w e e n
Dmorphisms,
D is the
8
A'
a 2cell
is like
oplax
opKmorphisms
We
Kcategories
K in
[~,
CAT/~.
CAT[
except
if we t o o k
of K  m o r p h i s m s . is O , O ' , ~
~ B'
8,
where
¥
: O ~ [ and y'
: O' ~ ['
satisfy
(Io.18)
~ ' ~
A'
B
~
B'
~
A
A'
y
 10.5
of [ 1 2 ] § 3 . 5
n.t.'s;
instead
§§10.3
', B
is y,¥'
A
all
KoL ~ LoK b e t w e e n clubs that
form A
This
CAT/~ for C A T f * ~
FUN is a f u n c t o r ~ : A ~ A', and a m o r p h i s m
of
(i0.17)
while
below
Kmorphisms
for
where
Denote
for two
for
K,L in CAT/~ t h e m s e l v e s
application
with
is,
in this volume,
of the
K is that
and
CAT/~.
A further
CAT~
a distributive
a distributive
10.8
[~,
with
in [8]
lie
that
is that
in v i e w
categories
or w h e n
L
K ~ L for clubs
themselves
that
for m o n o i d a l
and
handy,
considering
A and A';
club
categories;
categories
structures is o f t e n
K is the
]~..,
O' @'
~
B
~
B'
251 There (10.19)
CAT/~
~ :
sending (To@,
is an e v i d e n t
K, ¢ to Ko@,
To8 ' , To~),
(@,e',~)
to
notationally
~ from
It is f i n a l l y
clear
n
So
(e,
: T ~ T and
in the
we have
no n e e d
clear
sense
e',~)
(y,y')
there is r e a l l y
It is m o r e o v e r
FUN,
that
: K ~ K' and
that
: @ ~ @, to
: (8,
@',~)
~
to d i s t i n g u i s h (10.19)
provides
(KoL) ~ ~ ~ K ~ ( L g ¢ ) ,
that
etc.
a 2adjunction
FUN(K~¢,~) m CAT/~ (K,< ¢,@ >),
(10.20) where
o.
on
T
sending
qoy,).
of CAT/~
an a c t i o n
× FUN ~ FUN sending
and
(noy,
2functor
denotes
the c o m m a
(10.21)
= { 6 , 1 } / { i , ~ }
in
as in
CAT/S,
object
dI (10.22)
~
{A,B}
~
{A',B'} .........
~{A,B' }
{¢,I} Thus
FUN is e x h i b i t e d An a c t i o n
is c l e a r l y
@':KoA'
the
(@,0',~):
same
~ A', m a k i n g
 category.
K?@ ~ @ of the
club
thing,
by
§3.5 of [12],
A and
A'
into
KoA
(10.23)
CAT/S
as a t e n s o r e d
K in CAT/S
as a c t i o n s
Kcategories,
on @ E FUN
@ : K o A ~ A and
together
with
an ~,
~ A
Ko~
KoA'
~ A'
,
@, making
of
clubmap
(~,~)
a Kfunctor
K ~
A ~ A'
over ~ w h e r e
Such has
an a c t i o n its
then
evident
is a
clubstructure.
252 To g i v e a w h e n 8,6' whose composites
are a l r e a d y
given
is to g i v e s u c h a K ~
w i t h d o and d I are the g i v e n c l u b  m a p s
K ~ {A',A'}
and K ~ {A,A}. S u c h an ~ has c o m p o n e n t s
(lo.24) aT[AI...An] w h i c h are to be n a t u r a l (3.18)
of [12]
: T ( ¢ A I . . . ¢ A n) ~
¢ T ( A 1 . . . A n)
in T and in the Ai, and the a x i o m s
(3.17),
become
(zo.25) aT(Sl...Sn)[A1...Am I
~T[ SI(A 1
• . .
Am1 ) . . ..Sn( .
T ( ~ S I [ A i • . .Aml ] "''a S n [ • . .Am] )",
.Am)]
(i0.26)
If now K is g i v e n by g e n e r a t o r s description
of the s t r u c t u r e
give ~ in t e r m s only to give natural
=
in the A i.
To a r e l a t i o n
as a c l u b  m a p
of t h e s e g e n e r a t o r s
aT
~T[A1...An]
Kfunctor
in the strict m o n o i d a l identifications axioms case,
expressing etc.)
@(@,~)
a T by
(10.25)
a T = a S.
of a T in T o n l y
one has
and
to be (10.26).
It r e m a i n s
to
for m o r p h i s m s
category = @(~,@) in
that,
when K
or for strict m o n o i d a l
functor
naturality
Similarly
First,
is f i n i s h e d .
categories
is a m o n o i d a l
shows h o w to
for T E P, and it is of c o u r s e
In this way one sees for i n s t a n c e for m o n o i d a l
(P~p,D,~), the
and r e l a t i o n s .
i m p o s e the a x i o m
the n a t u r a l i t y
f : T ~ S in 9, and one
either
K ~
One t h e n gets the g e n e r a l
TpS one m u s t
i m p o s e as a x i o m s
and r e l a t i o n s
in the u s u a l case;
sense
the a x i o m s
for s y m m e t r i c
@(@,~)
a
f r o m the
~ @(~,®)
monoidal
categories,
(not a s t r i c t one
in this case are the a:
is the club
same as the in the n o n  s t r i c t
functors,
monad
functors,
253
and o t h e r w e l l  k n o w n In the q:
(¢,~) ~
cases.
same way we see that
(@,B):
transformation
A ~ A',
B:
¢ ~ ~
we need
as we commonly
~ T ( A 1 . . . A n)
q T ( A 1 . . . A n)
.....~~T(A1...An) ~T[A1...A n]
(3.19)
of [12],
do for m o n o i d a l
natural
only
We conclude
by n o t i c i n g
some other
special
to the case
of clubs
in
CAT~q,
K then be a club
in
CAT/~.
a Kcategory each
A such that,
i, the functor
for each choice gory,
monoidal
is a c a t e g o r y
closed
A bearing
variance kind of I left
T in P;
that
seem to be
CAT~set O.
or perhaps
By a closed
Kcategory
FT = n say,
a monad,
were
symmetric
cartesian
and for adjoint
closed
category,
cate
etc.
is when a K  c a t e g o r y
or a comonad.
when K actually
whether
lies
for a club
in p a r t i c u l a r
the q u e s t i o n
closed
such a thing
the algebras
§1.2 above;
monoidal
we m e a n
in
CAT/~,
the
L of the m i x e d 
they are monadic
the canonical
over CAT.
K ~ L in this
faithful. This
announced of Day
category,
in [7 ] that,
open there
case was
Thus
like to check what
Kcategories
properties
for each T @ K, with
of the A. E A. $
I showed
objects
T ( A 1 . . . A i _ 1 Ai+l...~) : A ~ A has a right
blclosed
The reader may
for those
transformations.
10.9
Let
on the natural
~T[ A I ... A n]
T(qA1...nAn) i T(@AI...~A n) the a x i o m
impose
the axiom
(lO.27)T(%A1..¢An)
representing
for a K  n a t u r a l  t r a n s f o r m a t i o n
question
in [ii].
in [i]
can be a n s w e r e d
In fact,
by using
it is to clubs
seems to extend.
a Yoneda
K in
CAT/~
embedding, that
as
the work
254
For
for a K in CAT/~, we can e x t e n d
if A is a K  c a t e g o r y
Kstructure within
on A to one
isomorphism)
on B = IA°P,setl , and
if we ask
indeed
B to be a c l o s e d
ITI:
B n ~ B of T E K m u s t
be the
its r e a l i z a t i o n
ITI:
A n ~ A.
the a b o v e
lies
in C A T / ~ ;
strict
Lmap
K; the
strict
since
they
Kmap
K ~
the
embedding
that
arbitrary always
is so.
above
doctrine
associativity
for
pseudoKalgebra
above
extension
for a club
below;
does
extension
question
when
the
of the
Yoneda
that
work
not
~ of
since
K ~ K o I is
for an
K in say CAT~cat O. need
K
embedding,
L is f a i t h f u l
not
of
is a u n i q u e
Yoneda
of c o u r s e
~ Set
loose,
in Set c a u s e s either
with
For
I is
But Set,
be.
for a club
since
the
K in CAT~set,
account
 I have
not
thought
of strict
to be only
of this
as we shall
Set by an e q u i v a l e n t
colimits
lack
B = [A°P,get}
we t a k e
algebras,
or we r e p l a c e
associative
K ~
a Kcategory
is a l i t t l e
in g e n e r a l ;
do in I81
with
for then
Kan
there
(to
r: K ~ set.
colimits
pseudoalgebras
of w h e t h e r
Yoneda
and yet {l°P,set}
comparing
strictly
on ~ E K).
is a c l u b  m a p the
coincides
i.e.
left
under
on !; w h e n c e
sat, is c e r t a i n l y
Of c o u r s e
image
(We are u s i n g
D, or even
a Dcategory,
there
effect
on I,
the
or at any r a t e since
~ E L to the
L ~ B then
same
free K  c a t e g o r y Note
B to be {K°P,set}
if we t a k e
L ~ B taking
have
the Y o n e d a the
for
solves
uniquely
Kcategory;
the r e a l i z a t i o n
This
the
by
to some category
out
the
a
extent admitting
question
set has these. REFERENCES
[11
B.J.
Day,
On c l o s e d 137
12}
S. E i l e n b e r g
(1970), and
G.M.
calculus, I3}
J.W.
Gray,
The
categories
Kelly,
Jour.
99
Lecture
Notes
in Math.
138. A generalization
of A l s e b r a
categorical
in Math.
of f u n c t o r s ,
3 (1966),
comprehension
(1969),
242312.
of the
functorial
366375.
scheme,
Lecture
Notes
255
[4]
J.R. Isbell, Math. Reviews
[ 5]
G.M. Kelly, Manyvariable Notes
[6]
G.M. Kelly, An abstract Math.
[7]
in Math.
281 (1972),
calculus.
I., Lecture
66I05.~
approach to coherence,
Lecture Notes in
106147.
G.M. Kelly, A cutelimination
theorem,
Lecture Notes in Math.
196213.
G.M. Kelly, Coherence theorems distributive
[9]
functorial
281 (1972),
281 (1972), [8]
44 #278 (1972).
laws,
G.M. Kelly and S. Mac Lane,
for lax algebras
and for
in this volume. Coherence
in closed categories,
Jour. Put e and Applied A l~ebra i (1971), [i0]
G.M. Kelly and S. Mac Lane, Closed coherence transformation,
[ii]
for a natural
Lecture Notes in Math.
G.M. Kelly and R. Street
(Editors),
97140.
Abstracts
281 (1972),
of the Sydney
Categgr~ ' Theory Seminar 1972, mimeographed by School of Mathematics, 1972;
second printing
editor at his present [12]
G.M. Kelly and R. Street,
128.
(originally
Univ. of New South Wales,
(1973) available
from either
address).
Review of the elements of 2categories,
in this volume. [13]
A. Kock, Limit monads Preprint
[14]
M.L. Laplaza, Math.
[15]
M.L. Laplaza,
in categories,
Series 1967/68 No. 6 (1967).
Coherence
for distributivity,
281 (1972),
F.W. Lawvere,
A new result of coherence
Ordinal
[17]
G. Lewis,
Coherence
in
for distributivity,
281 (1972),
sums and equational
Notes in Math.
Lecture Notes
2965.
Lecture Notes in Math. [16]
Aarhus Univ. Mat. Inst.
80 (1969),
doctrines,
Lecture
141155.
for a closed functor,
281 (1972), 148195.
214235.
Lecture Notes in Math.
256
[18]
G. Lewis,
Coherence
for a closed functor,
New South Wales, [19]
1974.
S. Mac Lane, Natural associativity Rice Univ.
[20] V. Z~berlein,
Ph.D. Thesis,
Studies
and commutativity,
49 (1963),
2846.
Doktrinen auf 2Kategorien,
(Math. Inst. der Univ.
Z~rich,
manuscript
1973).
Univ. of
DOCTRINAL
ADJUNCTION by
G, M.
This
paper deals
with
are categories with structure; isolated
observations
Kelly
adjunctions
its purpose
in the
n~:
f
I u:
is to unify
literature,
A ~ A'where A,A'
and
simplify
various
at the same time e x t e n d i n g
them
widely. (a)
First,
left adjoint (b) if f~u that
~ I.
tensor
others,
the forgetful
of which from
§§5.1 and
sufficient
5.2),
condition
and A' be symmetric merely (i.e.
in CAT but commute
class
morphisms
the best
(Ab, 0) to
(Set,
w h i c h was not
that a
in Sym Mon CAT,
with the c a n o n i c a l
f(X)
in natural
of c a r t e s i a n
known
tensor
that
the
case
~ fX @ fY
~ X + X, so examples
f
closed
but there
are
is that where
u is
of mine
to this
products;
the a d j u n c t i o n
and finally symmetric
and
×).
itself d i r e c t e d
closed,
closed,
f(X~'Y)
of an old t h e o r e m
for f to preserve monoidal
that
of topoi,
perhaps
One can dig out from the proof (131
result
in general
and let
h o w often
a whole
by geometric
functor
sense
let A ~ A' ~ Set,
products:
perhaps
it is not
in the
Yet it is s t r i k i n g
is p r o v i d e d
innum e r a b l e
categories,
in CAT,
products"
instance,
= A × A.
is the c l a s s i c a l
colimits.
an a d j u n c t i o n tensor
For
preserve
examples
there
If A and A' are m o n o i d a l
is m e r e l y
that u(A) does
f preserves
f "preserves
and fl'
and trivially,
question,
namely f~u
a
that A lie not
that u be normal
monoidal
functors
A ~ Set
and A' ~ Set). (c) with
Two t h e o r e m s
structure:
Theorem
of Street 9 of ~SJ
deal w i t h a d j u n c t i o n s asserts
for c a t e g o r i e s
that the left adjoint
of a
258
monadfunctor
is an o p  m o n a d  f u n c t o r ,
and T h e o r e m i of [9] asserts
that the left adJoint of a lax natural t r a n s f o r m a t i o n is an oplax natural transformation. (d)
Let V be a symmetric m o n o i d a l
be a m o n o i d a l m o n a d equalizers,
(= c o m m u t a t i v e monad)
the c a t e g o r y
"internalhom"
closed category and let T on V.
Then p r o v i d e d V admits
VT of T  a l g e b r a s is closed,
sense of [2]
in the original
(cf. Kock 71); it has been shown to have a
tensor product, m a k i n g it m o n o i d a l closed,
only when it is cocomplete;
and c o c o m p l e t e n e s s has only been d e m o n s t r a t e d under highly r e s t r i c t i v e h y p o t h e s e s on V and T. certain adjunctions
In this context Wolff [ii]
VT ~ VT'
has looked at
(arising in fact from a d i s t r i b u t i v e law),
and shown them by direct calculations to be closed adjunctions. the right a p p r o a c h here is to change universes, tensor product,
r e c o v e r i n g the m i s s i n g
and then to use the easier "monoidal" methods;
might also be of value to have a simple criterion, of internalhom, (e) conditions
Perhaps
but it
d i r e c t l y in terms
for an a d j u n c t i o n to be closed.
Finally there is Day's result Ill giving sufficient for a full r e f l e c t i v e
subcategory A of a b i c l o s e d m o n o i d a l
category A' to admit itself a b i c l o s e d m o n o i d a l
structure,
in such a
way that the a d j ~ n c t i o n f~~u, where u is now the inclusion,
becomes
a monoidal adjunction. It turns out that there are some simple general results that illuminate all of the above situations.
We start in §i with a doctrine
D, which could be on any 2category at all, but which we take to be a doctrine on
CAT,
purely because the n o m e n c l a t u r e is there more vivid.
(For what we need here about d o c t r i n e s we refer to [6] above in this volume, e s p e c i a l l y
53.5.)
We suppose that A and A' are Dcategories,
and that we are given an a d j u n c t i o n q,s:
f~u:
A ~ A' in CAT.
Our
first result is that there is a b i j e c t i o n between enrichments of u to a Dfunctor
(u,~) and enrichments of f to an o p  D  f u n c t o r
(f,f').
259
Our second result is that if the a d j u n c t i o n f~u admits enrichment to an a d j u n c t i o n above ~', so that
(f,~)~(u,~) (f,~) and
enrichment ~ is given, ~' is an isomorphism,
in DCAT,
then ~ is the inverse of the
(f,f') are strong; m o r e o v e r that if the
(u,~) has a left adjoint
in DCAT p r e c i s e l y when
the left adjoint then being
(f,~) where ~ and f'
are inverse; and finally that if the enrichment ~ is given,
(f,f) has a
right adjolnt in D  C A T precisely when it is strong, the right adjoint being
(u,~) where ~ corresponds by our first result to the inverse 7'
of f. The first result encompasses
the two theorems of Street in (c)
above  for the second of these the relevant a suitable CAT/A.
2category is not CAT but
The second result gives a proper answer to the
q u e s t i o n raised in (b) above: is to say that it is strong,
to say that
f preserves tensor p r o d u c t s
so that this happens precisely when the
a d j u n c t i o n is one in Mon CAT, which can also be expressed as a condition on u, here r e p r e s e n t e d by its generators u: UO :
uA@'uB ~ u(A@B) and
I' ~ ul (cf. §10.8 of [4] above in this volume)
This shows how
wide of the m a r k the sufficient c o n d i t i o n given in (b) above really is: that A and A' should be closed,
or even symmetric,
is totally irrele
vant; as for the n o r m a l i t y of u, we shall see that it is a consequence of the a d j u n c t i o n ' s lying in Mon CAT, and not an independent c o n d i t i o n at all.
The o b s e r v a t i o n
(a) that left adjoints preserve colimits can
also be seen as a trivial case of this second result,
by taking D as
the d o c t r i n e whose algebras are categorieswithcolimits: functor gives a c o m p a r i s o n of colimits, Dfunctor,
any
and is hence canonically a
so that the a d j u n c t i o n is a l w a y s in DCAT and f is always
strong, which here means c o l i m i t  p r e s e r v i n g . In §2 we look at the case where D is the d o c t r i n e whose a l g e b r a s are m o n o i d a l categories, but supposing A,A' to be a c t u a l l y m o n o i d a l closed.
Then the giving of G:
giving of a certain u:
u[A,B]
uA®'uB ~ u(A®B) ~ [uA,uB]'
is equivalent to the
where the square brackets
260
are the internalhoms;
~
and the conditions
on u, u
0
for the a d j u n c t i o n
A
to be m o n o i d a l
translate
into conditions
equivalent
to the n o r m a l i t y
not);
on ~ is i n d e p e n d e n t
that
"sufficient
condition"
on f, fo (namely on f and fo. on the
that
Now,
conditions
level,
products,
in (d) above.
(Recall
closed
natural
transformations.)
precise
conditions.
here
to be a closed
functor
interest
conditions entirely the
even in the absence
"simple
criterion"
categories,
functor,
proo~ would
and
desired
monoidal
similarly
for
still
be in terms
of n o n  m o n o i d a l
closed
and leave
of §i is m e r e l y
to suggest
and
categories
it at that
the
of
of these has not been developed,
ad hoc proof
results
into
is that they are indeed
an ideal
the theory
the c o n d i t i o n s
they are still
closed
or
is the
are e x p r e s s e d
one,
the
= closed
The answer
an independent
of the general
conditions
for m o n o i d a l
I suppose
in v i e w of the m a r g i n a l
translate
we can ask whether
that
= monoidal
but
the
so as to provide
functor
give
Similarly
on u ° is
are closed
a g a i n how m i x e d  u p
(b) above).
that
for the a d j u n c t i o n
proDstructures;
(showing
the c a t e g o r i e s
they be isomorphisms)
of the tensor
merely
of u (whether
g i v e n in
however,
internalhom
on u and u °. That
I
 the role
the right
conditions. Finally
in §3 we r e t u r n
turn to the a n a l o g u e
of Day's
A to be a full r e f l e c t i v e the counit
~:
Dstructure
to A too, Our results
shows
the enrichment condition;
By "roughly" structure
I mean
we show that
satisfied
Dstructure
it is possible
f must
is e s s e n t i a l l y
D and
suppose we suppose
be strong;
unique
to give a
the a d j u n c t i o n
to one
this at once
if it exists,
the c o n d i t i o n
only
So we now
or e q u i v a l e n t l y
enriching
in §i show that
doctrine
Now only A' is ~ i v e n as a
under w h i ch
that we may have
on A (axioms
get an honest
of A',
at the same time
in DCAT.
of a general
in (e) above.
fu ~ I to be the identity.
we seek conditions
a necessary
result
case
subcategory
Dcategory;
that
to the
is r o u g h l y
and gives sufficient.
to make do with a p s e u d o  D to w i t h i n
in two cases:
isomorphism).
for any D,
We can
if we a c t u a l l y
261
ask f to be strict condition); "the
and for c e r t a i n
same thing"
"flexible"; monoidal reduces
(accepting
one.
then,
our c o n d i t i o n
This
shows
is irrelevant;
D, where
The
for strict m o n o i d a l
to Day's.
bielosedness
correspondingly
"flexible"
as an honest
that
case,
the
since u must
by our
imposes
of [i],
upon h i m s e l f
I.I
is in fact no r e s t r i c t i o n
We refer h e a v i l y
results
on a d j u n c t i o n We suppose
Everything
CAT.
were n: n,c:
f~u:
and
above
for all
A ~ A'
DA
that
pairs
terminology
of the
of the
We further
of natural
on some
from [6]
u, which
we write
suppose
explained.
2category.
2category
in question,
as if the
2category
with actions
given
transformations
~
A'
Df
,
u,f' n
an a d j u n c t i o n
as in ~
A
5'
DA' ........
n'
We record
Secondly,
introduction
otherwise
DA
u
([6]
one;
both for its
A and A' are Dcategories,
u
which are mates
to normal
not
D = (D,m,j)
A
DA' .......
a "monoidal"
in CAT.
n
Du
First,
if A' is.
in this volume,
simplicity,
DA' ~ A'.
We c o n s i d e r (i.i)
to [6]
for n o t a t i o n a l
DA ~ A and n':
result.
and it
The m a i n results
given a doctrine
We suppose
paragraph
is
at all.
we say is quite i n d e p e n d e n t
and so, m e r e l y
or b i c l o s e d
is
In the
sufficient,
Day's
§2, the r e s t r i c t i o n
in the first
i.
and
is e s s e n t i a l l y
that A is closed
Day
about
categories
is not.
is n e c e s s a r y
we just get as a bonus be normal
D for m o n o i d a l
two things
necessary
a pseudoDalgebra
categories
the result
stronger
f
~
A'
•
n v
§2.2)
under
Proposition
the a d j u n c t i o n s
2.1 the value
Df~Du
and
of f' in terms
f4u. of u, to wit
262
n (1.2)
I
DA
~
DA'
~
A
~
DA'
A
 A'
1
n'
Denote by I, II the axioms for (u,[) to be a Dfunctor, (3.17) and
(3.18) of [6] with f r e p l a c e d by u.
Denote by I', II' the
c o r r e s p o n d i n g axioms for (f,~') to be an o p  D  f u n e t o r ~ t h e and
(3.18) of [6]
namely
same (3.17)
with } r e v e r s e d in sense and r e  n a m e d }', and with
A,A' interchanged. Le~ma i.i
Axiom I is equivalent
Proof.
Proposition
2.5 of [6}
to I' and II t_~o II' shows that the mates of the left
squares of the left sides of I and II are identities; P r o p o s i t i o n 2.4 of [6} is D}';
shows that the mate of the left square of the right side of I P r o p o s i t i o n 2.2 of [61
T h e o r e m 1.2
There is a b ijection b e t w e e n enrichments of u to a D
functor U = (u,[): F'
completes the proof.
= (f,}'):
A ~ A' and enrichments
A' ~ A.
of f to an o p  D  f u n c t o r
T h i s b i j e c t i o n is given by takin$ u and f' too
be mates" under the adjunct ions DfJDu and f 1.2
Now
{u.
consider a further natural t r a n s f o r m a t i o n f as in DA'
(1.3)
Df
n T
~
A
~
DA
f
~ n
A.
2~
L e t us are
call
I",
satisfied,
marion
s:
II" the
the
condi$ion
F U ~ i is,
(1.4)
axioms
by
DA
DA'
for
(3.19)
A'
=
(f,~)
c:
to be a D  f u n c t o r
F.
If t h e s e
fu ~ i to be a D  n a t u r a l
transfor
o f [6],
~ A
,n......
n'
for
~
n
DA
I
DA'
~A
i
Df DA We get Du,
an equivalent
which
to the
Df~Du a n d identity
thing
natural
process
to say t h a t
i~i.
A
DA
condition b y p a s t i n g
is an i n v e r t i b l e
same
~
n
The mate
by
we p a s s
D4 to e a c h
(2.1)
to m a t e s
of t h e r i g h t
transformation
and
of n. Df;
side
side a l o n g
(2.2) under of
the m a t e
n
of [6]:
(1.4)
n

,,
~
of the
A
~
f DA
which
in v i e w o f
(1.2)
n
above
(1.5)
c a n be r e w r i t t e n
as
DA
DA' ......
n'

A'
f
edge
it c o m e s
is t h e n t h e
Df DA
the
the a d j u n c t i o n s
DA'
"
~A.
~ A.
left
side
is
264
We conclude that,
if F = (f,~) is a Dfunctor,
and only if f'.f = i, the identity of n. Df.
shows that n:
An exactly
if
similar argument
i ~ UF is Dnatural if and only if f.f' = i.
Ii" are equivalent
Proposition
then ~ is Dnatural
Since i",
to I',II' when ~ and ~' are inverse, we have:
1.3
Given ~ and ~', and hence the m a t e u of the latter,
n and ~ constitute an a d ~ u n g t i o n in DCAT between Dfunctors
U = (u,~)
and F = (f,~) if and only if (i)
(u,~) is a Dfunctor;
(ii)
(f,f) is a Dfunctor;
(iii)
~ and ~' are m u t u a l l y inverse.
Moreover (i')
and
(i) is e q u i v a l e n t to
(f,~')
is an opDfunctor;
and (i') and (ii) are equivalent
in the presence of (iii).
Immediate consequences are:
Theorem 1.4
In order that a D  f u n c t o r U = (u,~):
A ~ A' have
a left adjoint F = (f,~) i~n DCAT, it is necessary and sufficient that f i_~n CAT, and that the ~' $iven by
u have the left ad~oint 
isomorphism.
T h e o r e m 1.5
Then f = f,1 , and F is n e c e s s a r i l y strong.
In order that a D  f u n c t o r F = (f,~):
a right adjoint U = (u,u) i~n DCAT,
D
A' ~ A have
it is n e c e s s a r y and sufficient that
f have t h e risht ad$oint u in CAT, and that F be strong. mate of f' = ~i , in the sense of (i.I).
1.3
(1.2) be an
w
Then u is the
D
We leave the reader to formulate the obvious dual theorems
o b t a i n e d by r e p l a c i n g the doctrine D by the opposite doctrine D*, where D*A = (DA°P)°P; he will get in
this way theorems about adjunction in the
2category of Dcategories and opDfunctors; adjoint that must be strong; and so on.
here it is the right
265
2.
2.1
The m o n o i d a l and closed cases
Now let D be either the d o c t r i n e whose algebras are m o n o i d a l
categories,
or else that
categories.
whose algebras are symmetric m o n o i d a l
These d o c t r i n e s arise from clubs in
CAT/~,
and it follows
from §10.8 of [4] above in this volume that a Dfunctor is the same thing as a m o n o i d a l case may be. u:
functor or a symmetric m o n o i d a l functor,
In both cases u is determined by its components
uA @' uB ~ u(A@B) and u°:
([2] page 473 in the m o n o i d a l
I' ~ ul, subject to the usual axioms case, with the extra axiom of [2] page
513 in the symmetric m o n o i d a l case). of [4]
as the
It further follows from §10.8
that u is an i s o m o r p h i s m precisely when u,u ° are isomorphisms.
Similarly a Dnatural t r a n s f o r m a t i o n is, in both cases, a m o n o i d a l natural transformation. Theorems 1.4 and 1.5 t h e r e f o r e apply to an a d j u n e t i o n ( f , ~ , f o ) _ ~ (u,~,u o) either in Mon CAT or in intermediary of an o p  m o n o i d a l
functor
Sym Mon CAT,
(f,~, fo,).
T h e o r e m 1.4 is now that f' and fo, be isomorphisms;
via the
The c o n d i t i o n in that in T h e o r e m 1.5
is now that f and fo be isomorphisms.
There is no d i f f e r e n c e between
the n o n  s y m m e t r i c and symmetric cases;
it is just automatic
that u
satisfies the extra axiom in the symmetric case if and only if f or f' does.
We get the explicit e x p r e s s i o n s
for fo,:
for f':
f(X@'Y) ~ fX @ fY and
fl' ~ I in terms of u and u ° from (1.2), giving D its value We find for fo, and f' r e s p e c t i v e l y
in terms of the a p p r o p r i a t e club. the composites (2.1)
fl'
~
ful
~
I;
fu °
(2.2)
f(x~,~)
~
f(~)
f(ufX~,ufY)
fX@fY.
fu(fX@fY)
f~
S
266
Given a m o n o i d a l functor U = (u,u,u°): A(I,):
A ~ Set and V' for A'(I',):
monoidalfunctor
structures
([2]
A ~ A', write V for
A' ~ Set with their canonical
page 504).
There is a canonical
monoidal natural t r a n s f o r m a t i o n V ~ V'U with A  c o m p o n e n t
A(I,A)
~
A'(ul,ui)
UlA
A'(I',uA), A'(u°,I)
as shown in [2] page 510; we call U normal if this is a natural i s o m o r p h i s m V ~ V ' U (we do not require it, as we did in [2], to be actually an equality).
P r o p o s i t i o n 2.1
If we use the a d j u n c t i o n f ~ u
U is normal p r e c i s e l y when
we find at once:
(2.1) is an isomorphism.
In particular normality of U is a n e c e s s a r y c o n d i t i o n for the existence inn Mon CAT.
°f an a d j u n c t i o n F ~ U
One can similarly t r a n s f o r m the c o n d i t i o n that
(2.2) be an isomorphism,
but I see no special m e a n i n g in the t h u s  t r a n s f o r m e d condition,
and omit
it.
2.2
Now suppose that the m o n o i d a l
categories A and A' are closed,
so that OB and @'Y have right adjoints of giving u:
[B,]
and [Y,]'
Then instead
UA @' uB ~ u(A~B) we may equally well give its mate under
the adjunctions
 ® B   ~ [ B ,  ] a n d @'uB~[uB,]';
w h i c h is a natural
transformation ^
(2.3)
u:
u[B,C]
.......
~
[uB,uC] '
It is shown on page 487 of [2] that the m o n o i d a l  f u n c t o r axioms for u,u ° on page 473 of [2] translate into the c l o s e d  f u n c t o r axioms for ~,u ° on page 434 of [2]; and similarly for m o n o i d a l natural t r a n s f o r m a t ions and closed natural transformations. We leave the reader to verify that the c o n d i t i o n that an i s o m o r p h i s m translates (2.4)
u[fY,C]
(2.2) be
into the condition that
......
[ufY,uC] ' 
~
In,l]
[Y,uC] '
267
be an isomorphism.
Moreover,
if f is given, and c o r r e s p o n d s to f,
then the c o n d i t i o n that f be an i s o m o r p h i s m translates into the condition that
(2.5)
[Y,uC],
uf[Y,uC]'
............^
n be an isomorphism.
~
u[fY,fuC]
uf ^
O
So ( u , u , u )
~
u[fY,C]
u[l,~]
has a left adjoint when u does if and
only if (2.1) and (2.4) are isomorphlsms;
and
(f,~,fo) has a right
adJoint when f does if and only if fo and
(2.5) are Isomorphisms.
We omit the proofs of these r e d u c t i o n s p r e c i s e l y because we are now going to give direct proofs of the last two statements that apply even to nonmonoidal closed categories. given in the introduction.
Our motive for doing this was
As the reader will see, the proofs that
follow largely parallel those of §I, and would doubtless be best seen in the context of p r o  D  s t r u c t u r e s ,
if the theory of these had been
worked out.
2.~
Our d e f i n i t i o n of (nonmonoidal)
tial m o d i f i c a t i o n of that of [2]; [ , ]: L:
closed category is an inessen
a category A, a functor
A °p x A ~ A, an object I of A, and natural t r a n s f o r m a t i o n s
[B,C]
~ [A,B],[A,C]],
being an isomorphism;
j:
I ~ [A,A],
i:
A ~ [I,A], this last
s u b j e c t e d to the axioms CCI  CC4 of [2] page
429, together with the axiom that the A(A,B) ~ A(I,[A,B])
induced by j
be an isomorphism. We suppose given as before an a d j u n c t i o n ~,e:
f~u:
A ~ A'
in CAT, but now we suppose A and A' to be closed, and not n e c e s s a r i l y monoidal.
It is still the case that there is a b i j e c t i o n b e t w e e n
m o r p h i s m s u°:
I' ~ ul and m o r p h i s m s
in terms of u ° by (2.1): f~u.
fo,:
fl' ~ I, where fo, is given
this is of course immediate by the a d j u n c t i o n
But in place of the b i J e c t i o n b e t w e e n natural t r a n s f o r m a t i o n s
u and natural t r a n s f o r m a t i o n s
f' that we had in the m o n o i d a l case, we
get s o m e t h i n g more complicated.
Precisely,
natural t r a n s f o r m a t i o n s
^
u as in (2.3) are in b i j e c t i o n with natural t r a n s f o r m a t i o n s
268
u1:
u[fY,C]
~" [ Y , u C ] ' , ^
u I being given i n terms of u as the composite ( 2 . 4 ) ; and n a t u r a l ^
transformations
f:
flY,Z]' ~ [fY,fZ]
are in b i j e c t i o n w i t h natural
transformations f1:
[Y,uC]'~u[fY,C],
fl being given in terms of } as the composite
(2.5).
To see that
^
u~~u I and f F~f I are indeed bijections, and
(2.5) in 2dimenslenal
we have only to write
(2.4)
form as, respectively,
(2.6) A°PxA
..... [..x. ]
>
op A'°PxA
A
LU ~A'°P×A
i
~ A'°PxA ' l×u
~
A',
[ , ]'
(2.7) [
A'°PxA '
A'°PxA
, ]'
..........................
>
i
>
A'
~ A'
A'°PxA a,, A°PxA
I
[ ,1
f°Pxl ^
e x h i b i t i n g u I as the m a t e of u under certain adjunctions, and fl as the mate of f under others, in the sense of [6]
§2.2.
A closed functor is of course said to have an adjoint one in the 2category of closed categories, natural transformations.
The m a i n result of this section,
and e x t e n d i n g the assertions
T h e o r e m 2.2
closed functors,
of §2.2 above,
if it has and closed suggested by
is:
A closed functor U = ( u , ^u , uo) : A
~ A' has a left
adjoint F = (f,~,fo) if and only if u has the left adjoint f and m o r e o v e r fo, and Ul, given by (2.1) and
(2.4) respectively,
are
269 ^
isomorphisms. t__~ofl = Ul
Then fo is the inverse of fo, and f corresponds by (2.5)
i ^
A closed functor F = (f,f,fo):
A' ~ A has a right adjoint
^
U = (u,u,u °) if and only if f has the right ad~oint u and m o r e o v e r fo a n d fl' the latter ~iven by (2.5), are isomorphisms.
Then u
O
^
c o r r e s p o n d s by (2.1) t o fo, = fol, and u corresponds by (2.4) t__o_o u I = fl I . ^
Proof
^
Take all the data u , u ° , f , f ° as given, with the mates
ul,f°',fl of the first three, without supposing yet that the closedfunctor axioms CF1CF3 of E2] page 434 are n e c e s s a r i l y satisfied by U or by F.
This last in no way prevents us from
the c o m p o s i t e s FU = ~ = (¢,
m e c h a n i c a l l y forming
,¢o) and UF = ~ = (@,~,~o) as on t2] page
434. Our first o b s e r v a t i o n is that q:
i ~ ~ and s:
¢ ~ 1 satisfy
the axioms CN1 and CN2 for closed natural transformations,
on page
441 of 121, if and only if fo is inverse to fo, and fl to u I.
In
fact s satisfies CN2 if and only if flu1 = I; n satisfies CN2 if and only if ulf I = l; s satisfies CNI if and only if fo,fo = I; and n satisfies CN1 if and only if fofo,
= 1.
We prove only the first of these assertions, to the reader.
Express
flul = i by pasting
leaving the others
(2.6) on top of (2.7).
In
^
the r e s u l t i n g d i a g r a m the 2cells f and u have a common edge, and the result of pasting them along this edge is just ~. 2cells are the triangles n °p x I, q, i x s. of these by pasting on an s °p x i
The other three
Get rid of the first two
and an s to cancel them out, as in
(2.1) and (2.2) of ~61; the e x p r e s s i o n of flu1 = i is now that ~ with a I x s pasted on is equal to the identity of uE s °p × I and an s pasted on in suitable places. c in 2  d i m e n s i o n a l form.
, ](fop x i) with an But this is just CN2 for
27O
This establishes the "only if" part of both the a s s e r t i o n s of the theorem.
For the ~if" part of the first assertion, we use the
p r e s c r i p t i o n of the t h e o r e m to define f and fo; and it remains to show ^
that F = (f,f,fo)
satisfies the c l o s e d  f u n c t o r axioms;
similarly for
the "if" part of the second assertion. We shall show here that F (resp. U) satisfies the a x i o m CF3 of [2] page 434 when U (resp. F) does, and when the hypothesis
is
verified that u I and fl are inverse; we leave the easier axioms CFI, CF2 to the reader.
pl,p2:
Write the two legs of CF3, say for ~, as
~[Y,Z]
~
[~[X,Yi,[~X,~Z]];
we are now for simplicity w r i t i n g [Y,Z]
for [Y,Z]'
We similarly use
pl,P2 for the two legs of CF3 for U or for F. We do not, a priori, n:
know CF3 for ~; but we do know that
i ~ Y satisfies CN2, and by an easy d i a g r a m  f i l l i n g  i n argument
we get that (2.8)
[Y,Z]
=
~[Y,Z]
n is independent of i.
~
[~[X,Y],
[~X,¢Z]]
Pi Now give @ its value uf, and express the Pi
for ~ in terms of those for U and for F, as is done in the big d i a g r a m on page 435 of [2], which we shall call A. First suppose we know CF3 for U. (2.9)
~
Then
(2.8) and A give that
uPi
[Y,Z] ~ uf[Y,Z] ~ u[ f[ X,Y] ,[ fX,fZ]]
[uf[X,Y],
ulfX,fZ]]* [uf[X,Yl,[ufX,ufZ]]
ll,;l is independent of i.
But since
(2.4) is the i s o m o r p h i s m Ul, the
^
m o r p h l s m s u and ¿l,u] cancelled:
in (2.9) are b o t h c o r e t r a e t i o n s and can be
the composite of the first two m o r p h i s m s in (2.9) is
already independent of i. diagram
This
implies that the top leg of the
271
fuP i
fq .............
flY,Z]
~
fu[ f{ x,Y] ,I fx, fz] ]
fuf[ Y,Z]
[ fiX,Y]
f[ Y,Z]
'Pi is independent triangle
of i.
But the square
commutes
by one of the adjunctionequations;
of i, which
by naturality, hencePi
and the
is independent
is CF3 for F.
Now suppose We conclude
,[fX,fZ]]
instead
that we know CF3 for F and seek it for U.
from (2.8) and A that
(2.10) [Y,Z]
n ; uf[Y,Z]
uf >
Pi u[fY,fZ]~
[u[ fX,fY] ,[ufX,ufZ]] I [ uf,l] luf[ X,Y] ,l ufX,ufZ] ]
is independent
of i.
Write this for Z = uC, compose
[l,[l,uc]]
: [uf[X,Y] , [ufX,ufuC]]
naturality
to move this last morphism
on the end with
~ [uf[X,Y] ,[ufX,uC]] , and use ^
in the process
it becomes
first three morphisms isomorphism (2.11)
u[l',e]
:
back through u[ fY,fuC]
in the thustransformed
fl of (2.5) and can be removed.
u[fY,C]~
[uf,l]
and Pi;
~ u[fY,C] ; so that the composite
(2.10)
form the
What we now have is that
[u[fX,fY],lufX,uC]]
> [uflX,Y],[ufX,uC]] ^
Pi is independent ule,l] l~,l]
: u[B,C]
of i.
[ uf, i] Now set Y = uB, compose with
+ u[fuB,C]
: uf[X,uB],ufX,uC]]
naturality
on the front,
* [X,uB],lufX,uC]]on
to move the first morphism
that the last three morphisms [f1,1],
we have that
and with the end.
through the second,
then give an instance
Using and observing
of the isomorphism
272
(2.12)
u[B,C]
[ u[ fX,B] ,[ ufX,uC] ]
~Pi
is i n d e p e n d e n t
of i.
the c o m m u t a t i v i t y
The
"extraordinary"
naturality
of Pi in A gives
of Do
u[ B,C]
t
...........
[ u[ A,BI ,[ uA,uC] I
[ u[ fuA,B] ,[ u f u A , u C ] ]
~
[ u[ A,BI ,[ u f u A , u C l 1 ;
[l,[u¢,l]] since by but
(2.12)
the top leg is i n d e p e n d e n t
[l,[u¢,ljjis
retraction
a coretraction
by one of the a d j u n c t i o n
is in d e p e n d e n t
of i, which
3.
~.i
cancellable,
equations.
of
Reflective
§I to g e n e r a l i z e
So once 2category
special (3.1)
again we are on w h i c h
ua b e i n g a
the Pi on the left
subqatq$ories
Day's
result
D, and use the
in [i] .
given an a d j u n c t i o n
D acts,
doctrine
n,e:
but now we suppose
f~u:
A ~ A' in
that we are
in the
case where fu = I
The results this were
and
being
CAY;
~ = !:
fu ~ i.
independent
then
subcategory
adjunction
equations f~ = i:
of the
2category,
(3.1) may be e x p r e s s e d
reflective
(3.2)
Thus
leg;
is CF 3 for U.
We return n o w to the case of an a r b i t r a r y
results
the
and hence
of i, so is the b o t t o m
of A', with
((2.1)
and
f ~ fuf;
by c a l l i n g
inclusion
(2.2)
of [6]
~u = i:
we again w r i t e A a full
u and r e f l e x l o n above)
u ~ ufu.
as if
f.
now become
The
273
We suppose a c t i o n n': giving
A', but not A, to be g i v e n with a Dstructure,
DA' ~ A'.
By an e n r i c h m e n t
of a D  s t r u c t u r e
enriching way that
to A, with a c t i o n
of u and f to Dfunctors n,~ provide
By T h e o r e m
an a d j u n c t i o n
1.5 to give
DA'
n:
U = (u,u) F~U
situation
DA ~ A say,
we m e a n the and the
and F = (f,~)
in such a
in DCAT.
an enrichment
to A and to give an i s o m o r p h i s m (3.3)
of this
with
is to give
a Dstructure
f as in
n'
A'
f Df
f
DA
~
.....
A
n
that m a k e s at all, n.Df
a strong
it is u n i q u e
~ fn',
determined (3.4)
of (f,~)
while
Dfunctor.
If such an e n r i c h m e n t
to w i t h i n a suitable
(3.1)
gives
Df.
isomorphism;
for
exists
(3.3) gives
Du = i, so that n is e f f e c t i v e l y
by
n ~ f.n'.Du. Now consider
the c o m p o s i t e
(3.5)
DA'
....~.
DA'
i If an enrichment which
enrichment
to exist
D is the d o c t r i n e single
exists,
is the i d e n t i t y
condition
~
A'
~
n'
by
(3.5)
is isomorphic
(3.2).
is that
Thus
(3.5)
for m o n o i d a l
by
(3.3)
a necessary
this
to n. Df.D~,
condition
be an isomorphism.
categories,
A.
f
In the
clearly
for an case where
reduces
to the
274
(3.6)
f(n®'n):
f(X®'Y) ~ f ( u f X ® ' u f Y )
and Day [I] has shown that in this case
is an isomorphism; (3.6) is also sufficient.
H o w e v e r the i n v e r t i b i l i t y of (3.5) can hardly be sufficient for a general D; if D were the d o c t r i n e
for strict m o n o i d a l categories,
the
i n v e r t i b i l i t y of (3.5) would still reduce to (3.6), but the m o n o i d a l structure c o n s t r u c t e d by Day on A, given by A@B = f(uA@'uB), would not in general be strict even though that on A' was strict. the Introduction, "flexibility"
As we said in
the doctrine for m o n o i d a l categories has a certain
 an ability to absorb
isomorphisms
 which the doctrine
for strict m o n o i d a ! categories lacks. It turns out that the i n v e r t i b i ! i t y of (3.5) is n e c e s s a r y even for the existence of a p s e u d o  e n r i c h m e n t ; sufficient.
To be precise,
and that for this it is also
let D' be the d o c t r i n e whose a l g e b r a s are
the n o r m a l i z e d pseudo D  a l ~ e b r a s in the sense of Street [i0] volume.
§2 in this
That is to say, a D'category differs from a Dcategory in
that the a s s o c i a t i v i t y axiom for the action is satisfied only to within a (prescribed) axiom h o w e v e r nose.
isomorphism,
subjected to suitable axioms;
(this is what "normalized" means)
T h e n a D '  f u n c t o r resp.
calls a lax h o m o m o r p h i s m [resp. algebras.
is what Street
strict homomorphism]
every strict D  f u n c t o r is a f o r t i o r i d o c t r i n e  m a p p:
is satisfied on the
strict D'functor]
Since every Dcategory is a f o r t i o r i
the unitary
of p s e u d o  D 
a D'category,
a strict D'functor,
and
there is a
D' ~ D; and it is shown in my paper [5] b e l o w in this
volume, which also Justifies the other remarks above, that p is a retraction. ~.2 T~orem
We can now give the m a i n result of this section. ~.!
Let the r e f l e x i o n ~,~:f~u:
A ~ A' be ~ i v e n as above,
and let A' be a Dcategory, hence also a D'category.
Then the
r e f l e x i o n admits a D '  e n r i c h m e n t if and only if (3.5) is an isomorphism.
275
Proof pA':
First, the necessity. D'A' ~ DA'.
Consider the composite of (3.5) with
By the 2naturality of p, the m o r p h i s m pA' can be
m o v e d past the triangle Dn in (3.5), turning the latter into D'n. But then pA' composed w i t h n': A'.
DA' ~ A' is just the action of D' on
Thus the composite of (3.5) with pA' is just the analogue of (3.5)
with D' r e p l a c i n g D. a D'enrichment
We already know that this must be invertible
is to exist.
But then (3.5) itself must be invertible,
since pA' is a retraction. We turn to the sufficiency. n:
Guided by (3.4), we define
DA ~ A by
(3.7)
n = f.n'.Du.
We can then write (3.8)
¢ = f.n'.Dn
(3.5) as an i s o m o r p h i s m :
fn' ~ n.Df,
and we define the ~ of (3.3) to be the inverse of ~. We check that n satisfies the unitary law for an action. have n.jA = f.n'.Du.JA = f.n'.jA'.u
by (3.7) by the n a t u r a l i t y of j
=
fu
by the unitary law for n'
=
1
by (3.1).
As regards the associative law, we have n.mA = f . n ' . D u . m A
by (3.7)
= f.n,.mA,.D2u
by the n a t u r a l i t y of m
= f.n,.Dn,.D2u
by the a s s o c i a t i v e law for n';
while n.Dn : n . D f . D n ' . D 2 u Therefore the i s o m o r p h i s m
if
by (3.7).
We
276
(3.9)
U = ¢.Dn'.D2u:
f.n'.Dn'.D2u ~ n.Df.Dn'.D2u
is an i s o m o r p h i s m (3.10) ~:
n.mA = n.Dn,
and we define v to be
i
It remains to verify that n and v constitute a n o r m a l i z e d p s e u d o  a c t i o n of D on A, and that f and ~ c o n s t i t u t e a "lax h o m o m o r p h i s m of p s e u d o  D  a l g e b r a s " in Street's language. we have a D '  a c t i o n on A and a strong D '  f u n c t o r whenqe the desired D '  e n r i c h m e n t
rather different:
(5),
(f,f):
A' ~ A,
follows by T h e o r e m 1.5.
The axioms to be v e r i f i e d are the axioms §2 for v, and the axioms
Then in terms of D',
(6) of [I0]
(i),
§2 for f.
(2),
The notations are
Street's i is our j, his c is both our m and our n,
as well as our n'; his ~ is i in our case
(normality);
his e is our v,
and also our v' which is 1 (A' being a honest Dcategory); is our f. inverses,
(3) of [I0]
and his ¢f
In v e r i f y i n g these axioms we replace v and ~ by their ~ and % respectively,
Axiom (i) reads
inverting the arrows accordingly.
~.jDA = i.
~.jDA
But
=
¢. Dn'.D2u.
=
%. jA'. n'.Du
This will therefore follow from axiom
¢.jA'
Axiom (2) reads p.DjA p.DjA
jDA
by (3.9) by the n a t u r a l i t y of j.
(5), which reads ¢.jA'
: i.
But
=
f.n'. D~,JA'
by (3.8)
=
f.n'.jA'.n
by the 2naturality of j
=
fn
by the unitary axiom for n'
=
i
by (3.2).
=
i,
But
=
¢.Dn' .D2u.DJA
by (3.9)
=
¢.Dn' .DJA' .Du
by the n a t u r a l i t y of j
=
¢.Du
by the unitary axiom for n'
=
f.n' .D~.Du
by (3.8)
277
=
i
since nu = I by (3.2).
This leaves us with axioms reduces
to the second,
the vertical vertical
composite
composite
via (3.9).
¢.mA'.
(6).
The first of these
For the left side of axiom
of n.D~ with ~.DmA,
of ~.D2n with ~.mDA.
and the right
Now using
=
n.D¢.D2n'.D3u;
u.DmA
=
¢.Dn'.D2u.DmA
=
¢.Dn,.DmA,.D3u
by the naturality
=
¢.Dn,.D2n,.D3u
by the associativity
~.D2n
=
~.D2f.D2n,.D3u
by (3.7);
~.mDA
=
¢.Dn'.D2u.mDA
=
¢.mA,.D2n,.D3 u
and the right So axiom
n.D¢
side is the vertical
(3) follows
It remains
by the naturality
(6) is the vertical
from axiom
then to verify
axiom
by (3.7) and
¢.Dn'
=
f.n'.D~.Dn'
by (3.8);
~.D2f
=
f.n'.Dn.Dn'.D2u.D2f
by (3.9) and
¢.mA'
=
f.n'.D~.mA'
by (3.8)
f.n,.mA,.D2n
by the 2naturality
f.n,.Dn,.D2n
by the assoclativity
composite
with D2n'.D3u.
Now
f.n'.Du.Df.Dn'.D2~
The vertical
of n.D¢ with
of ~.D2f with
=
=
axiom for n';
of m.
(6) on composing (6).
side is the
of m
composite
composite
(3) is
(3.9)
n.D~
But the left side of axiom ¢.Dn',
(3)and
of the first two of these
(3.8);
(3.8);
of m axiom for n'.
is indeed equal to that
of the second two, both being D2A
DA
~" DA' i
Dn'
~ D A ' ~ i
A' n'
= f
A.
278 This
completes
3.7
the proof.
As we have
we can a c t u a l l y
said in the Introduction,
get a Denrichment.
not m e r e l y
an i s o m o r p h i s m
F = (f,f)
to be not m e r e l y
strict.
This may
in nature, extremely
useful
Theorem
~.2
strong
[5]
first
an a r t i f i c i a l
be by T h e o r e m
case;
show,
(unpublished)
is that where
and
~,s:
f~u:A
Then the r e f l e x i o n
w i t h F strict
(3.5)
Proof must
if and o n l y ~ i f f.n'.Du.Df
For the necessity, have
f.n'
fn = i by
= n. Df.
I have
found
it
related
to
~ A' be g i v e n as above, admits
is the i d e n t i t y
a Denrichment
(which includes
we are to have
(3.3)
is n. Df.Dn,
with f = i, so that we
which
is the identity
since
is c o n t a i n e d
case we have
in the proof of T h e o r e m
¢ = 1 and ~ = i, or equally
3.1,
since
f = 1 and
D We
said towards
is a retraction; D = D'
that
the end of is, there
with pq = 1.
doctrinemap.
§3.1 that
the d o c t r i n e  m a p
is a 2natural
In general
q to be a
q with pq = l, we call the
doctr i n e
D the doctrine
D' is flexible;
and we also
show that D is
flexible
if D is of the form K °, where
K is a club
in
in the or that
sense
whose
discrete
of [4]
monoidal
below,
we show that
JKJ of objects
§10.3 above.
for symmetric
functoroperations
club
[5]
suppose
D flexible.
CAT/set)
In the paper
transformation
we cannot
If we can find a d o c t r i n e  m a p
p: D' ~ D
doctr i n e
in
the
= f.n').
So (3.5)
sufficiency
in the present
q:
1.5) but
(3.2).
The
v=l.
to a s k i n g
it formally:
Let the r e f l e x i o n
that
is
it o f t e n arises
considerations
and let A' be a Dcategory.
assertion
but
in w h i c h
(3.5)
which corresponds
(as it must
b e l o w will
in various
To state
The
there are two cases
but an identity;
seem rather
as the paper
coherence.
0
Thus
the club
categories,
are g e n e r a t e d
for any
CAT/~
(or even
is a free d i s c r e t e for m o n o i d a l
is flexible,
freely by @ and I; the
since club
club
categories, the for strict
279 m o n o i d a l categories axioms like ®(®,~)
escapes the result
since ® and I are subjected to
= ®(~,®).
By a p p l y i n g the theorem b e l o w in the case where D is replaced by D', we see that T h e o r e m 3.1 remains true even when A' is originally given as a D '  c a t e g o r y rather than a Dcategory.
T h e o r e m ~.~
Let the r e f l e x i o n q,c:
above~ and let A' be a Dcate~ory~ r e f l e x i o n admits a Denrichment
f~u:
A ~ A' be ~iven as
where D is flexible.
Then the
if and only if (3.5) is an
isomorphism. Proof
The necessity was shown in §3.1.
For the sufficiency,
use T h e o r e m 3.1 to get a D'enrichment. qCAT:
D'CAT ~ DCAT
(cf. [6]
first
Then apply the 2functor
§3.6 above)
to get a Denrichment.
The only point at issue is whether the Dstructure A' now has is that it started with;
this is ensured by pq = i. D
REFERENCES
l 1]
B.Day
(=B.J.Day), A r e f l e c t i o n t h e o r e m for closed categories, Jour.
[2]
Pure and Applied A l g e b r a 2 (1972), iii.
S. E i l e n b e r g and G.M. Kelly, Categorical Algebra
I3]
Closed categories,
(La Jolla 1965),
G.M. Kelly, On clubs and doctrines,
[5]
G.M. Kelly, Coherence theorems distributive laws,
in this volume.
for lax algebras and for
in this volume.
G.M. Kelly and R. Street, Review of the elements of 2categories,
[7]
Lecture Notes in
106 (1969), 166177.
[4]
[6]
Conf. on
S p r i n g e r  V e r l a g 1966.
G.M. Kelly, A d j u n c t i o n for enriched categories, Math.
Proc.
in this volume.
A. Kock, Monads on symmetric m o n o i d a l closed categories, Arch. Math.
21 (1970),
iI0.
280
[8]
R. Street, The formal theory of monads, Jour. Pure and Applied Algebra 2 (1972), 149168.
[9]
R. Street, Two constructions Top. et G$om.
[ io]
on lax functors,
Diff. XIII 3 (1972),
Cahiers de
217264.
R. Street, Fibrations and Yoneda's lemma in a 2category, in this volume.
[ 11]
H. Wolff, Commutative distributive laws, to appear in Jour. Australian Math. Soc.
COHERENCE
THEOREMS
FOR LAX ALGEBRAS
AND FOR DISTRIBUTIVE
LAWS
by
G.M.
i.i
We p r o v e
two m a i n
o n Cat.
(For t h e m o s t
Kelly
results.
basic
First,
notions
about
let D be a d o c t r i n e , doctrines,
see [13]
say
above,
,
familiarity (honest)
with which
algebras
are
we a s s u m e . )
Let D
the lax a l g e b r a s
be t h a t
for D,
doctrine
in t h e
sense
whose of [24]
,
above.
Since
a Dalgebra
is a f o r t i o r i
a D algebra,
there
is a
,
doctrlnemap
[Cat,Cat]
s:
D
~ D.
of e n d o  2  f u n c t o r s ,
doctrinemap)
such that
sB = 1 a n d n h = I.
Cat, "full by
the
existence,
of a m o r p h i s m
h:
D ~ D
sh = I, a n d o f a 2  c e l l
Borrowing
the
terminology
in the
n:
2category
(not a
1 ~ hs
usual
s u c h that
in the
2category W
we m a y
say t h a t
reflective
"pseudo",
this
the
same
case
this
result,
along with
For law between for
natural
second
ones
examples
hopelessly
Moreover, true,
rare
[i0]
result,
at the d o c t r i n e for c a t e g o r i e s , with
actions
Dactlon
is g i v e n
the
structure
some
doctrine
D
onto
"lax"
the
above
n o w n is a n i s o m o r p h i s m ;
of e n d o  2  f u n c t o r s .
let p:
original
provided
for
if we r e p l a c e
we
shall also
D ' D ~ DD'
or r a t h e r
in the
of D
Part
prove,
so
of
has
above.
category
algebra
but
some r e f i n e m e n t s
two d o c t r i n e s
strict
s as a " r e f l e x l o n "
s is a n e q u i v a l e n c e
in m y p a p e r our
D.
is s t i l l
in t h i s
been applied
exhibits
subobject"
that
law;
We prove
.
sense
level while
both
a pseudodistributive
of B e c k
(just
[2[
are r a r e
in
as A @ B = B ~ A is
A @ B a B @ A is c o m m o n ) .
of D a n d o f D',
of a ( n o n  s t r i c t ! ) Were
be a d i s t r i b u t i v e
and
for w h i c h
D'morphlsm,
the d i s t r i b u t i v e
A the
is a n
law a strict
one,
282
DD' would
itself be a doctrine,
with a D  a c t i o n D'morphism; case there
and a D'action,
this
pseudodistributive rate a D algebra; whose
reflective
subobject
monoidal
again,
monads
strictly
ones are easily monoidal
distinguished, closed
two symmetric
is a special for a more formulate principles
results
  now more
One
let us now call
doctrine,
monoidal
there
an e q u i v a l e n c e
of
(although
the e n r i c h e d
those
similarity
~
that he called
this
D'
the
monads.
Once
for symmetric if D is any
[19],
In p a r t i c u l a r
D'.
In that
who considers
@ and @ together
it applies
case
our
a category
with
with a
(A@B)~(A@C).
lax
the present
very
general,
situation. proofs
have
in §2 and §3 below,
of this
on a
but p s e u d q  c o m m u t a t i v e
Then
applies.
particular
case of s o m e t h i n g
collected
are rare,
of our two m a i n results
lax and a less
monads
is a p s e u d o  d i s t r i b u t i v e  l a w
result
structures
A®(B@C)
on coherence
diagrams.
onto a full
often called m o n o i d a l
that of Laplaza
as we have
strict
i = ii ~ DD'.
to be a strong
such is the d o c t r i n e
The rest comme n t s
of D
found.
this:
but is at any
is jj':
doctrines
"distributivity" The
among
category,
and the above
contains
In our
as the unique
it is a c t u a l l y
the D  a c t i o n
when D too is taken to be this result
i ~ D
In that
D'morphism.
categories;
D'D ~ DD',
and that
~ DD'.
~ DD'
commutative
pseudocommutative p:
D
a doctrine,
D
:
be categories
of Beck { 2 }.
s is a r e f l e x i o n
if we require
still not a strict)
commutative
with j
that
DD';
s:
would
being a strict
s:
is no longer
so we can define
is once again
symmetric
DD'
composition
Kock [14]
situation
be a d o c t r i n e  m a p
case,
Our result
endo2functors
the D  a c t i o n
is the classical
would clearly
D morphism
and its algebras
introduction
problems,
relating
that each
the d o c t r i n e s
I have not a t t e m p t e d in common only
distinct.
to general
the r e l a t i o n
about
to
some general
and are otherwise
is devoted
including
stated and results
suggests
between
the c o m m u t a t i v i t y
such of
2~
1,2
To be told,
for some d o c t r i n e and the 2cells implicitly. that
D
situations,
(along with the
strict m o r p h i s m s
between
the
strict m o r p h i s m s )
By the "coherence
of d e t e r m i n i n g
soluti o n
as in the above
of this
I understand
D
problem"
explicitly
is o f t e n beyond
a complete
are the a l g e b r a s of these,
is to be given D
for D* I u n d e r s t a n d
from this
information.
our powers.
or partial
what
solution
primarily
A complete
By a "coherence of this problem:
theorem" a
,
result
w h i c h tells
us
something
of Cat.
For instance,
functor
D; or even that
as a full r e f l e c t i v e
that
at least
about
it is e q u i v a l e n t
it contains
subobject.
D
~ua endo2functor
to some k n o w n
endo2
a k n ow n D, in some d e f i n i t e
Results
about
certain
diagrams
commuting
m a y be part of such a theorem,
or may be among its
important
consequences;
can,
seen as c o n s t i t u t i n g
but
the essence
I shall return First,
I want
simplified (a)
to observe
in various
The word
"monad".
the case
(b)
such as the
on Cat.
Aindexed
family
with
monoidal
like
internalhom,
An e q u a t i o n a l
Cat/A.
2cells,
themselves
point.
to be r e p l a c e d
structural
categories,
structure
closed
V.
structure
are the
functors
are all
subjected
functors
if one considers may be borne are a l g e b r a s
as in of m i x e d
only
by a for a d o c t r i n e
of d o c t r i n e s
But then too there
an E E LCat,CatJ the
by
involved.
and these
i ~ E satisfying
diagrams.
for a mere m o n a d when,
are
be
has been over
may have
It is easy to conceive
"lax doctrines";
E 2 ~ E and unit certain
the
closed
of categories,
2category
of c o m m u t i n g
some e q u a t i o n a l
on Cat when
no longer
theorem.
of its polemical
(= 2monad)
VCat for a symmetric m o n o i d a l things
sake
One takes m u c h too n a r r o w a view
doctrines
on the
for the
endowed
I believe,
the last p a r a g r a p h
but they are the a l g e b r a s
of symmetric
variance,
of a c o h e r e n c e
to the m a t t e r
that
ways
for a d o c t r i n e
covariant;
later
"doctrine"
Categories
algebras
such results
way,
"doctrine
on are
with m u l t i p l i c a t i o n axioms"
to socalled
only to within
"coherence
axioms";
2~
such an E is an a l g e b r a
~Cat,CatJ.
Clearly
a mere monad) reasonable
(c) term
some d o c t r i n e
we should
take D
on an a r b i t r a r y
to suppose
such as being
for
locally
2category
K; a l t h o u g h
(or p o s s i b l y
it m a y be
some of the good p r o p e r t i e s
of
Cat,
presentable.
It is r e a s o n a b l e "coherence
on the
to be a d o c t r i n e
2category
that K has
D
theorem"
 and p r o b a b l y for a result
very
which,
common having
 to use the found out
,
something
about
these a l g e b r a s Mac Lane's
D
from a k n o w l e d g e
and deduces
original
something
coherence
saying that the d o c t r i n e  m a p functors,
where
and D that
D
gives a complete which
further
determination
consequence
that
symmetricmonoidally although takes
it for granted
case,
and in one other,
"Kategorien
1973").I
equiv a l e n c e
s:
to a D  a l g e b r a pursue would
D
has not
people's
D
symmetric
monoidal
to a strict
to refine
that
one.
statement
But
in fact
it has
category IThis
is
in this
It is a special
is true,
has been
and he
(at the O b e r w o l f a c h
every
conference
of such an
D algebra
is D *  e q u i v a l e n t
been discussed;
nor do we
out for investigation,
of what a c o h e r e n c e
known
Beck 13]
the e q u i v a l e n c e
than
P
the
is well
in print.
the existence
 but it cries
categories
but this a s s e r t i o n
more
as
of endo2
This
of fractions.
in ~71,
instance,
is Po for a club
result).
to my k n o w l e d g e
ideal
categories.
(since
in the proof
~ D implies
For
monoidal
Lane's
How g e n e r a l l y
it in this paper be m a n y
of D
it asserts
error
them.
may be i n t e r p r e t e d
for symmetric
any precise
of Isbell
about
goes back to
~ D is an e q u i v a l e n c e
via c a t e g o r i e s
by its author; a radical
every
E23]
monoidal
and proceeds
case of the a s s e r t i o n
discovered
D
by Mac
equivalent
I do not recall
withdrawn
s:
symmetric
is wholly d e t e r m i n e d
useful
theorem
is the d o c t r i n e
for strict
of its algebras,
theorem
and
should be.
285
1,3
We return now to the m a t t e r of c o m m u t i n g diagrams.
In some
cases we know at once from the d e s c r i p t i o n of the D algebras that D is Co for some club C, and that C is given by specified generators and relations; given there.
see §I and §I0 of [9]
above, and the further references
The w o r d  p r o b l e m involved in finding the objects of
C
is t y p i c a l l y easy  often indeed they are freely generated  while that involved in finding the m o r p h i s m s is typically m u c h harder. It is a matter of finding the category g e n e r a t e d by a certain graph subject to certain relations  a g e n e r a l i z a t i o n of the w o r d  p r o b l e m for monoids,
inasmuch as a category is a "monoid w i t h many objects".
Deciding when two words in the g e n e r a t o r s represent the same m o r p h i s m of C is d e c i d i n g which diagrams commute in C.
This is the most
common sense in which solving a coherence problem,
completely or
partially, may involve proving that certain diagrams commute.
The
typical example is that of symmetric m o n o i d a l closed categories,
the
club for which was d e t e r m i n e d in part by KellyMac Lane [ii] , and then more fully by Voreadu
([27],[28],[29]),
but is still not known
completely. (Other examples where the club is partially but not completely d e t e r m i n e d are: Vcategories,
a symmetric m o n o i d a l closed category V, two
two Vfunctors,
and a Vnatural t r a n s f o r m a t i o n
(Kelly
Mac Lane [12] ); two symmetric m o n o i d a l closed categories and a symmetric m o n o i d a l functor
(Lewis [ 21] ,[ 22] ); a category with two
symmetric m o n o i d a l structures ~ and @ along with a "distributivity" A®(B@C) ~ (A®B)@(A@C)
that is not required to be an isomorphism
(Laplaza [ 18] and [19]). d e t e r m i n e d are:
Some examples where the club is completely
a m o n o i d a l or symmetric m o n o l d a l category
(Mac Lane
[231); a category w i t h a t e n s o r  p r o d u c t and a n o n  i s o m o r p h i c a:
(A@B)®C ~ A®(B@C)
(Lapiaza [17]); a category with a monad
(Lawvere [20]; Lambek [16])
 the doctrine is just ~x; two symmetric
m o n o i d a l categories and a symmetric m o n o i d a l functor
(Lewis [21]
and
286
[ 22]).
I do not
of clubs
suggest
that
by their authors;
lastmentioned
result
but this
of Lewis
for it sets out to determine augmentation meant
r:
the above results is what
they amount
is so expressed,
C completely
C ~ Y is not
are expressed
faithful.
to.
in terms The
and indeed must
in a case where Finally,
this
be:
the
list
is not
to be exhaustive.) ,
One could of d e t e r m i n i n g
D*A for every
free D algebras. who
also regard
This
sets up generators
the d e t e r m i n a t i o n
object
A;
that
is,
of D
as the p r o b l e m
of d e t e r m i n i n g
is the view taken by Lambek
([15]
and [161),
for D *A , and who attacks
and relations
the
the
,
problem
of which diagrams
category.
in D A commute
To this end he has
Gentzen's
in place
knowledge
of C i m m e d i a t e l y
well at the level
of C.
"equigenerality
D A, and
Lambek's
studying
(cf.
When,
idea of setting
commutativity
for a club C, of
 for any
knowledge
of CoA,
and
techniques
those results
commutativity"
of
turn out
and some of them are
however,
up generators
of diagrams
of
the c o n s i d e r a t i o n
In fact,
implies
[8 ]§6).
is Co
the c u t  e l i m i n a t i o n
of " r: C ~ Y is faithful";
seen to be false
adaptation
complication
corresponding while
of the form
from a club,
gives
or not;
to be r e  p h r a s i n g s then
by Lambek,
of C is an u n n e c e s s a r y
this for any A, discrete work p e r f e c t l y
a brilliant
Yet when D
considered
D A = CoA
Lambek
introduced
work on cutelimination.
as it is in the examples
when A is a discrete
at this
D
does not
come
and r e l a t i o n s level,
for
m a y well
,
be the best way of getting D algebra
is a c a r t e s i a n
by Szabo
(]251
1.4
Thus
establishing
at D
.
closed
An example
category,
such results
w h i c h has been
investigated
and [ 26]). proving
diagrams
a coherence
commutative
theorem,
stated
in §I.i.
imply theorems
may be a tool
it; but it
some grip on D , as is shown by
On the other hand about
in
or even a way of stating
need not be the only way of g e t t i n g our results
is that where a
diagrams
I now point
commuting,
out that
at least
when
287
D,D'
come from clubs:
and imply them wholesale,
clubs are arbitrary. argument
In this way I believe
for calling
them
"coherence
P
CAT/~,
for a club ~ in
, and that
s,h,n all arise
one clinches
it is easy to see that
then D* is P * o from things
for another
change n o t a t i o n
and write
sh : i, s~ = I, n h = i. necessary
condition
in general ~S:
it always
P
If f,g:
~
T ~ S is a d i a g r a m
for it implies
It is however
is in the
"pseudo"
when S is in the image
of h, i.e.
in D
is sf = sg.
, a
This
is not
only that nsf = ~ s g , where
sufficient case,
such club
so that we may as p* P ~ , ~: i ~ hs,
D, h:
for its c o m m u t a t i v i t y
sufficient,
S ~ hsS.
s:
if D
in CAY/~;
*
well
the
theorems".
In the first of those results is Do
in so far as these
if ~S is an isomorphism,
and as it is in the when S belongs
"lax"
as
case
to the full
,
subcategory explicitly
~ of D
Since
and directly;
commutativity class
.
in the The
s is a clubmap,
sf can be c a l c u l a t e d
so we have an effective
of any d i a g r a m
in the
"pseudo"
test
case,
for the
and of a large
"lax" case. same
is true
in the
second
result
of §i.i,
in wh i c h
it is first
stated (the D  a c t i o n
If f,g:
T ~ S in D
with S in the full r e f l e c t i v e
then f = g if and only if sf = sg.
an a r b i t r a r y
in the form
D'morphism).
DoD',
subcategory
In the case where
the D  a c t i o n ,
is to be a stron~
D'morphism,
comes
in CAT/~ when D and D' do.
from a club
indicated
in §1.5 of [9];
Laplaza's
problem
distributivity
d:
of two
it is no longer
in the special symmetric
A~(B¢C)
~
and we can take it to be in ~ by p a s s i n g But
if we ask d to be an isomorphism,
its type neither
in ~, and ddl:
(A~B)¢(A®C)
in S nor in soP;
the d o c t r i n e
any sense we can at present
This
structures
to
and a
the type of d lies to the opposite
its inverse ~
is for the reasons
case c o r r e s p o n d i n g
monoidal
(A~B)¢(A~C),
the case that D
does not
come
give to that notion.
doctrine.
d I already
(A~B)¢(A~C)
in ~op,
has
has its type from a club
in
288
What we can do is to r e t u r n
to the club D
in the n o n  s t r o n g
,
case,
and consider
a model A
image of d happens which the image
(that
is, a D algebra)
to be an i s o m o r p h i s m
of n happens
(or more
in w h i c h
generally
to be a m o n o m o r p h i s m ) .
the
one in
Then although
,
sf = sg does not their
images
whose
objects
general
f,g:
f = g in D
in {A,A},
the
at the m o d e l  l e v e l
involving With
category"
So for
of [ 91
are
§9,
suitably
such a m o d e l A we have
for c o m m u t a t i v i t y
in D
of
 which m e an s
a
of any d i a g r a m
(in Laplaza's
case)
a
d but not d I.
this
said,
case the doctrines
and with
D and D'
mo~oidal
aategories,
original
result
which
[23],
includes
the o b s e r v a t i o n
that
are both the doctrine
is equivalent
so that
it is easy to see that our result
imply the e q u a l i t y
"rich e n d o  f u ~ c t o r
transformations.
T ~ S describable
diagram
, it does
are functors A n ~ A and whose m o r p h i s m s
natural
crlt~rion
imply
his theorem
as a map D
"distortion"
in [19]
P for symmetric
to ~ by Mac Lane's
s can be r eg a r d e d
s is L a p l a z a ' s
in L a p l a z a ' s
§4.
([19]
~ ~o~,
§2),
We give more
and that
details
below. 1.5
We end this
coherence
theorem
introduction
exemplified
by o b s e r v i n g
by the results
that the kind
of
of
§i.i
is pretty
result
in [231,
common. First,
if we take Mac Lane's
just at the a s s o c i a t i v i t y a:
(A~B)~C ~ AS(BSC)
part,
original
we see that the i n v e r t i b i l i t y
is not central
to the m a i n argument.
and look of
What
is
,
really p r o v e d
is that
s:
D
~ D is a reflexion,
where D is the
,
subcategory bracketed that
of the a p p r o p r i a t e
wholly
club R
from the right.
the r e f l e x i o n
result
T ~ S whenever
in that
by L a p l a z a
case.
in [17].
S lies
of the objects
It is only b e c a u s e
is in fact an equivalence.
even when a is not an isomorphism, for f = g:
, consisting
a is i n v e r t i b l e
It follows
sf = sg is n e c e s s a r y in D.
This
A total one r e q u i r e s
and sufficient
is a partial
more
work,
that,
and
coherence is g i v e n
2~
To get back to the partLal one, virtually
predictable,
of the present Identifying functor means h:
however:
I say that
it is
and in a sense automatic.
The m e t h o d
of proof
paper r e q u i r e s
a D functor
P with the set of natural
by h(1)
giving
= !, h(n)
h:
h2 ~ hl ~ h3, and this is of
course
axiom if it is to be a D functor: axiom for a.
but note that axioms"
similar
maps l:
situation,
but arises
A final say for m o n o i d a l monoidal
categories,
Mac Lane's
result
D
~ D.
But Mac
they are e~uivalent This
equivalence
pairs h: ([21]
with hardly was needed
variables
is not
an artificial
in the old, unsolved, closed
original
D' is not
gives
towards
But we can get a p a r t i a l
[61
from A, not
in
of the
form hA.
CAT/~ club
result
h.
Lewis
completely
for a ~eneral
the d i a g r a m s
they
above.
and c o n s i d e r
by a Cfunctor
in the m o n o i d a l
from A';
as e n d o  2  f u n c t o r s .
the end of §3 of [13]
connected
that
equivalence
to our D ; a l t h o u g h
the c o r r e s p o n d i n g
Observe
categories,
is an e q u i v a l e n c e
a similar
isomorphic
result, for strict
for m o n o i d a l
(P) cases were always rather special:
had c o d o m a i n
coherence
D' ~ D that
let C be a club
by E i l e n b e r g  K e l l y
only
category.
as d o c t r i n e s , and not m e r e l y
any effort.
coherence
If D : ~o is the d o c t r i n e
has d e t e r m i n e d
when C : P or N.
monoidal
A~I ~ A; this
A ~ A' of C  c a t e g o r i e s
and [221)
observation;
I and non
a doctrinemap
example,
at once to the
object
on Mac Lane's
is d i s c u s s e d
As a last
an
have a constant
Our first result
Lane's
has to satisfy
what
and if D' ~ No is that
gives
of e n d o  2  f u n c t o r s .
~Fedlct
nonsymmetric
categories.
of
"coherence
precisely
comment
components
the
I~A ~ A, r:
for a nonmonoidal,
a; the other (h,h)
h as a
we need
and this reduces
considerations
sh = i.
it to a D functor
is in itself a l i g h t w e i g h t
should be when we also
isomorphic
problem
This
Enriching
But
with
~0, we define
in p a r t i c u l a r
are then g i v e n by an easy induction.
pentagonal
D ~ D
numbers
= ~ ~ h(nl).
hn 0 hm ~ h(n+m);
h:
whose
commutatlvity
(N) and
they
C
symmetric
contained
lay in A'; and they always
A commutativity
criterion
for
such
290
diagrams only
from A means
algebra the
can be g i v e n
looking
on A = I and A'
free
algebra
it is i m m e d i a t e "unknown". l:
at once, not
at the w h o l e
= I where
on A = I and A' that
Since
its d o m a i n
h
is free
C ~ C is a m o d e l ,
rendering
for a g e n e r a l
there
C.
Having
club,
I is the unit = empty.
Let
is C itself, on the g i v e n
are u n i q u e
which
is the
category, this
while
C'
free
but
be h:
at
C ~ C';
is the
generators
strict
variables
and
Cfunctors
since
n,s
commutative C
n
• C
1
L
IC
C'
s
and having
n(~)
free
Ccategory
show
that
= ~.
But
on ~,
gives
s is in fact
embedded byh;
so that
Lewis
proves
incredibly
Since
a doctrine in the w i d e A:
A,B,
if and
this
sense:
sh = I.
of C' onto image
only
since
the
C is the
It is easy full
to
subcategory
C,
of h, the d i a g r a m
if sf = sg.
A lot
but m u c h m o r e
less
than
general,
very
cheap.
A method
section
on an a r b i t r a r y
L ~ K, not
L = I).
so that
for C = P or N, a d m i t t e d l y ;
2.
2,1
n = i;
requirement,
if S is in the
T ~ S in C' c o m m u t e s
and
last
a reflexion
f,g:
useful,
this
of c o n s t r u c t i n ~
is p u r e l y
necessarily
As far as p o s s i b l e
formal,
2category
an a c t i o n
a:
we take
(D,m,j)
K, and w e u n d e r s t a n d
DA~A
to be
"Dalgebra"
of D on a 2  f u n c t o r
on an o b j e c t we use
Dreflexions
a,b,
A of K(the etc.,
special
for a c t i o n s
case of D on
etc. We a s s u m e
and d o c t r i n e s
given
acquaintance in [13]
with
§3 above;
the
general
we r e c a l l
facts
about
in particular
monads that
the
291
free D  a l g e b r a DA has action mA: a:
D2A ~ DA, and that an action
DA ~ A is a strict Dmorphism.
We add a few enrichments
a p p r o p r i a t e to the doctrine case. First, (2.1)
Dy:
if y:
r ~ r':
Dr ~ Dr':
not only are Dr, Dr'
A ~ B is any 2cell, then in
DA ~ DB
strict Dmorphisms but Dy is a D2cell.
Next,
for a free Dalgebra DA and a Dalgebra B, there is an isomorphism not only of sets but of categories (2.2)
[L,K]
¢:
(A,B) ~ DAZ~,(DA,B).
(Here DAg@, has as objects the D  a l g e b r a s of domain L; its m o r p h i s m s are strict Dmorphisms,
and its 2cells are D2cells.
We recall
that we replace D~Z@, by DA£@ when we a l l o w all Dmorphisms as 1cells.)
In detail, S
r
(2.3)
@ sends
A
/ ~ y ~ B
where (2.4)
to
r v
A ~ y ~ " ~
B
=
A
B
S'
~ DA.//@B~B, S t
r * S
DA . .
~8
S
JA
(2.5)
DA
Dr
#e
,
B
DA~/~
=
~Dy
~'DB
• B. b
S v
Now c o n s i d e r an a r b i t r a r y D  m o r p h i s m G = (g,g):
(i.e. not n e c e s s a r i l y strict)
A ~ B, so that g is a 2cell a
(2.6)
DA
~ A
g
Dg
DB
~ B b
292
is a D2cell
Theorem 2,~
a (2.7)
DA
~A
Dg
DB
bB
. . . . . . . . . . .
,
all the edges now bein$ r e s a r d e d as Dmorphisms~
of which Dg,a,b
are strict. Proof
The D  n a t u r a l i t y axiom (3.19) of [13]
a s s o c i a t i v i t y axiom (3.18) of [13] Observe that,
for g reduces to the
for a D  m o r p h i s m
(g,g). D
since b.Dg = Cg by (2.5), we can also write
(2.7) as a D2cell
(2.8)
g:
Cg ~ Ga.
Observe further that the other axiom for a Dmorphism, axiom (3.17) of [13],
(2.9)
the identity
gives
A ~
DA
.............
jA
• B

A
II i
B.
~g g
2.__~2
Now,
in the situation of T h e o r e m 2.1, replace A by a free
D  a l g e b r a DA.
Define the strict D  m o r p h i s m WG and the D  2  c e l l
@G by: DA
(2.10)
D
A
B
=
D
~ DjA
~ B ¢g
Had we replaced DJA by iDA on the right side of this definition, should by (2.9) have got merely the identity. have
Since, however, we
we
293
JA DA
A
JA
i JDA DA
we can conclude
.... DJA
~ D2A
from (2.9), not that @G = i, but that DA G
(2.11)
DA~
A
B
A
i
jA
oAf
~G
Note in particular (2.12) Theorem
that, by (2.4),
~G = ¢(g.jA). 2.2
Dmorphisms
@G: YG ~ G is the coreflexion DA ~ B.
That is to say, if s:
Dmorphism and ~:
s ~ G is a D2cell,
(2.13)
s
~
=
i ~G
(2.14)
8.
DA ~ B is a strict
then
i~ G
B for a unique D2cell
of G into the strict
~a ExPlicitly,
~ = ¢(~.JA),
and B is the unique D2cell
satlsfyin@
TG
(2.15)
A
g
~ D A ~ ~ B jA
=
A
~
~ D ~ ~ B . jA
S
Proof
Since CG.JA = i by (2.11),
turn gives
S
(2.13)
implies
(2.14) by (2.4), proving the uniqueness
(2.15), which in of ~ satisfying
(2.13). Define
B therefore
value of B is b. D(~.jA).
by (2.14),
so that by (2.5) the explicit
Using this, the definition
the fact that mA. DJA = i, we get (2.13)
(2.10), and
if we compose with DJA the
294
diagram expressing
the Dnaturality
of a, to wit mA
mA D2A
D2A
~ DA
DA
Ds
DB
,~ B
s
g
DB
b
The naturality Proposition
(2.16)
2,3
of ~,~ is summed up in:
Let G be the composite
DA
• C
where H  (h,h),
~B
w E
s
H
t and s are strict,
t and s = ¢r for r:
Then
A~C.
G (2.17)
Proof
DA ~ ~ G G ~G
~G =
Corollary (2.18)
B
=
DC
DA
• B.
h
t
Dr
g.
by
DJA
=
t.h.Ds.DJA
=
t .~.
Dr
2.10) by (2.16)
by (2.4).
L,e~ G be the composite
2.4 DA
DC
rE
Dq
H
~ B;
t
then (2.19)
D
A
B ~G
Proof
=
DA
~B.
P D Dq
In this case the r of Proposition
~H
2.3 is jC.q.
t
[]
295
2.~
In [i0]
adjunction given
s~
§3 a b o v e
h:
B ~
we c o n s i d e r e d
the
problem
B' to a D  a d J u n c t i o n ,
in the
B' as a D  a l g e b r a .
enrichment
to exist;
Dmorphism;
A condition
when
it does,
and we c o n s i d e r e d
must
be
case
satisfied
s necessarily
in p a r t i c u l a r
of e n r i c h i n g sh = i, for the
becomes
the
an
a stron~
special
case w h e r e
s is to be strict. Here starting
point, For
that r:
we are A ~ B;
that
we a p p r o a c h in the
case
same
situation
where
B'
is a free
this
section,
we s u p p o s e
that
given
a strict
Dmorphism
s:
that
we are g i v e n
from a d i f f e r e n t Dalgebra
B is ~ i v e n DA ~ B,
DA.
as a D  a l g e b r a ;
say
s = Cr for
a Dmorphism
H = (h,~):
B ~ DA;
else
in o r d e r to get
and
we have
(2.20)
sH
=
i
as D  m o r p h i s m s . n:
the
1 ~ Hs
data
We a s k what
satisfying
sq = i and n h
for a D  r e f l e x i o n Write
which
Dmorphism, between
(mere)
by
and
C:
2cells
(2.4).
~:
¢(jA) ~:
then
by
On the o t h e r
hand
i:
By T h e o r e m
1 = Hs = G and
JA
so as to c o m p l e t e
Hs;
~ ¢(hr)
(2.21)
= I,
are
~ hr,
as
a D2cell the
B.
composite
is ¢(jA).
D2cells
D2cells
of DA onto
G for the
is ¢(hr)
we n e e d
2.2,
~G
= ¢(h.s.jA),
DA ~ DA is a strict
there
D2cells
in t u r n
(2.12)
is a b i j e c t i o n
C:
1 ~ YG;
in b i j e c t i o n
by
such
(2.2)
with
in
A
~ DA.
jA Using to w r i t e
(2.5)
YG and $G'
to w r i t e we
~ = ¢~,
see by T h e o r e m
and u s i n g
(2.17)
2.2 that
n is the
(with
t : i)
composite
296 B
(2.22)
of
DA
course
Dr
the
the b o t t o m
top
~ DB
leg
of
Since s~
this
=
s.mA.D~
=
b. DS.
=
b.D(s~)
=
¢(s~)
s~.
By
D~
Theorem
Summing 2.5
by
and with
D2cells
if a n d o n l y
n:
method
in o n l y widely
one
(2.23)
•
sn = i if a n d o n l y
if
up:
i ~ Hs a n d
s:
DA ~ B a n d H:
2cells
~:
B ~ DA with as in
jA ~ hr;
and
(2.22) s~ = i
G reflexion,
this,
is a n i m p o r t a n t
of the
and
and
we a l s o n e e d n h = i; I see no
and have
it v e r i f y
it by a n ad hoc
case.
in s i m i l a r
~ G = ¢(hr)
(2.5),
strict
sH = i, t h e r q is a b i j e c t i o n
way of expressing
There
= s by
sh = I, so
s is
Dmorphisms
i,,f, s~ = I.
in e a c h
b.Dr
two p r o b l e m s problems.
I = ¢(JA),
;
= i.
of c a t e g o r i e s ,
To get o u r d e s i r e d general
since
(2.20),
(2.5)
all thus Given
s = Cr s t r i c t between
hs
since
¢ is an i s o m o r p h i s m
= i.
is
l e g is I s i n c e m A . D j A
Now consider
sn
DA
special
case which,
studied
in this
It is t h a t w h e r e
this
A
is t h e
paper, Y G = I.
case where
~ JA
DA
while
it o c c u r s
does
occur
Since
297
co~nutes.
The
condition
if we n o w c h o o s e
~ = 1.
on us if we want n h n.jA
~h
= l;
= l; since ~ G . J A
is I~ = 1 by
(2.4).
= l; this
special
case
Theorem
2.6
If the d a t a
It m u s t and
this
3.
We are
applying
Theorem
that
going
(2.23)
gives
gives
~.jA
~ = 1 is not
is f o r c e d
= l; w h i c h
sufficient
ad hoc
verification.)
Summing
are
in T h e o r e m
and
if h . D r . h
it is a u t o m a t i c
What
going
~:
its
= ~. Dr o b t a i n e d satisfies
for
up this
and d e a l
It s a t i s f i e s
case
announced
Theorem
we a s s e r t e d
1.5 of [i0]
in §i.i
2.6.
in §i.i; Once
as it is in b o t h
the
by
So we are the a d j u n c t i o n
given,
however,
results
of §i.i,
above
that
any
So our w o r k i n g
with
D adJunctions
of our r e s u l t s
this m e t h o d
~ = i
as d o c t r i n e  m a p s
to be a D  a d j u n c t i o n .
by T h e o r e m
settin~
D
special
D morphism,
if m o r e o v e r sn = 1 and
by
s~ = 1.
our r e s u l t s
than
2.5,
1 ~ Hs w i t h
and D  m o r p h i s m s
to get m o r e
we can r e c o g n i z e
n
= I.
to a D  a d j u n c t i o n . not
as
to o b t a i n
2.5 and
s is a strict
requirement
case,
automatically
going
is a c t u a l l y
enriches
this
one D  2  c e l l
Dal~ebras
3.1
s4h
qhr
(2.11),
be the D  2  c e l l
= i if and only
seemingly
= 1 and
then
choice
satisfied
gives:
~h
~h
for
requires
there is at m o s t
i_~n (2.22),
hand t h i s
(Even l u t h i s
still
automatically
On the o t h e r
= i by
hr = jA, = I.
s~ = i is of c o u r s e
but
of p r o o f with
such a d j u n c t i o n
of our m e t h o d
of proof.
does
however,
require,
nonstrict
D morphisms,
is a
is that
when
all we
,
are
told
DAlg,
originally
about
of D  a l g e b r a s ,
Moreover
! here m e a n
an a c t i o n
of D
D
the u n k n o w n
strict
Alg,
on an o b j e c t
doctrine
D morphisms, in the n a r r o w of K,
not ,
However
in s a y i n g
we are also
given
that the
we are
given
forgetful
D
and
D
is the
D 2cells.
sense:
a D algebra
on a 2  f u n c t o r
Alg,,
fun ctor
from
Alg,
is
L ~ K.
I do m e a n D
2category
to
to K;
imply
that
and
it is
298
classical purpose allows
that all this determines
of this
section
us to r e c o g n i z e
~.2
once
"small h o m  c a t e g o r i e s "
from [4] these and
limits
As is well known,
is,
pullbacks
that
that
the
can then
and the cotensor
each K(A,B)
construct
2commaobjects
extends
2category
to 2cells).
as does Cat/A
The
Do
to doctrines
for a small
size of A.
Kan extension.
these
extensions,
category
property
category
It fails
the right
Kan e x t e n s i o n
extension
is V. However
E×K ~ K, sending
all those
and
E(TS,R)
K, in the
to DA,
D on Cat
here
~ E(T,V)
sense
IA = A ( a l o n g Moreover
of the
to CAT by
the p r o p e r t i e s What
to small
as tensor
category;
of
is
categories.
product
It and
only for certain
for some V, namely when then this right
that there
and s a t i s f y i n g
2cells).
of the form
of endo2functors.
composition
of R a l o n g S exists;
(ED)A = E(DA),
for 1cells
extend
[K,K]
extend
independently
categories
to be a closed
E acts on (D,A)
small
with
on Cat
new in the club case.
2category
S,R in E is it the case that
things
~ is the a r r o w
common d o c t r i n e s
we do not pursue
D takes
E for the
monoidal
laws
in K, u s i n g
itself has these properties,
certainly
which add n o t h i n g
We write
identity
A E K.
(the u n i v e r s a l
we can always
However
for us is that
i as identity.
of the
on CAT;
For that m a t t e r
is a strict
for n E Cat,
club D do so  one can form RoA
right
important
K ~ Cat;
for a set A.
It is true that many canonically
Cat
(b) that
K(A,):
where
0 ~ i; and these are then
We recall
limits;
comma objects
[ ~,AI
Kan
is a 2functor
the a s s o c i a t i v i t y
and
with the c o r r e s p o n d i n g we have
.
is a small
all small
In,A]
of D
K has
i.e. C a t  c o m p l e t e .
products
product
knowledge
2category
by the r e p r e s e n t a b l e s
cotensor
we
the simple m a c h i n e r y
(a) that K admits
are p r e s e r v e d
(c) that K admits
The
for all that
 that
this means
isomorphism.
from the implicit
and that K is 2complete, that
to within
is to introduce D morphisms
We now assume
category;
D
2natura!ly
299
(3.1)
K(DA,B)
where
{A,B}
is
m E(D,{A,B}),
the
right
Kan extension
of
B:
I ~ K along
A:
I ~ K.
We know e x p l i c i t l y what this is, namely for C in K (3.2)
{A,B}C = [K(C,A),B],
the c o t e n s o r product of K(C,A) C Cat with B ~ K.
The counlt of the
a d j u n c t i o n (3.1) is a 2natural e v a l u a t i o n e:
{A,B}A ~ B, and there
is similarly a 2natural unit d:
We have formally the
D ~ {A,DA}.
same kinds of things as we have in a closed category, a 2natural m u l t i p l i c a t i o n H: l:
in p a r t i c u l a r
{B,C}{A,B} ~ {A,C} and a unit
I ~ {A,A}, which make K into an Ecategory.
(That the u n d e r l y i n g
2category of this E  c a t e g o r y really is K itself is immediate, K(1,{A,B})
since
m K(A,B) by (3.i).)
It is m o r e o v e r easy to check that the. Evalued e n d o m o r p h i s m object unit
{A,A}, with its m u l t i p l i c a t i o n p:
i:
i ~ {A,A}, is a doctrine;
action a:
DA ~ A corresponds under
{A,A}{A,A} ~ {A,A} and its
and that for any doctrine D, an (3.1) precisely to a d o c t r i n e  m a p
D ~ {A,A}. 3.3
We now write K' for the functor 2category [2,Kl
the 2category which is like the [~,K~
of [9]
and K" for
§2.2 above except that
its m o r p h i s m s are not lax n a t u r a l t r a n s f o r m a t i o n s but oplax ones; is what we called FUN in [9] is d e s c r i b e d there
§I0.8 above in the case K = CAT, and it
(in the context of just such c o n s i d e r a t i o n s as
follow, but there specialized to the club case). So K' and K" have the same objects, namely m o r p h i s m s f:
A ~ B in K; a typical m o r p h i s m in K" is a triple
form u
(3.3)
A
it
......
f
c~ '~ B
~A'
If ' " r~B'
~T
(u,v,a) of the
3OO a typical q:
2cell
(u,v,a) ~ (~,~,~)
is a pair of 2cells
v ~ ~ in K satisfying the obvious
~; and the various compositions
conditions
are by pasting.
of (3.3) commute;
u ~ ~,
with respect
to ~ and
The morphisms
are those of K" in which e = i, and are thus just pairs making the outside
p:
of K'
(u,v)
its 2cells are those in K"
between these morphisms. There are evident actions being the restriction (D,(u,v,~)) everything
of the second;
goes to (Du,Dv,Da), in sight,
and 2cells of E. right adjoints;
namely
(D,f) goes to Df,
and in general D is applied to
with appropriate
arrangements
we have 2natural
K'(Df,f')
~ E(D,[f,f'l),
(3.5)
K"(Df,f')
~ E(D,< f,f') ),
isomorphisms
is the pullback ~0
(3.6)
[f,f']
b {A,A'}
{l,f,}
{B,B'}
~{A,B'} {f,l}
and < f,f'>
is the comma object
DO (3.7)
also for the 1cells
Just as in the case of K, these actions have
(3.4)
where [f,f'J
of E on K' and on K", the first
< f.f,)
P{A.A'}
I{l,f'}
{B,B ~}
~{A,B'}
{f,1} of course there is a canonical map
;
301
(3.8) with 61
s:
[ f,f']
~0 ~ = 60,
are j o i n t l y Thus
same
kind
observe this
~is
= 81,
K'
that [ f , ~ that
D ~ (f,f)
= I; it is a m o n o m o r p h i s m
and K" b e c o m e
of f o r m a l
case
~e
since
60
and
like K, w i t h
the
monomorphic.
6o,
BI,
~0,
are
~I,
taking
doctrines;
it is easy
s are
which
under
when
just
In p a r t i c u l a r ,
for any d o c t r i n e
corresponds
of D on f in K",
Ecategories,
properties. and ( f,f}
Moreover k:
~ < f,f')
to
see in
all d o c t r i n e  m a p s .
D in E,
(3.5)
written
and
f' = f, we
a doctrinemap
to an a c t i o n
(a,b,~):
Df ~
f
out as
a
(3.9)
A
DA r_
Df
f
f DB
is e a s i l y
seen
, ,,
to a m o u n t
~B
to a c t i o n s
on A and B p t o g e t h e r
wlth anenrichment
F = (f,~):
The
A ~ B.
action
a:
DA ~ A and b:
a:
~ of
DB ~ B of D
f to a D  m o r p h i s m
DA ~ A i t s e l f
corresponds
to the
doctrinemap (3.10)
and
D
similarly
Dmap; there
the
to a c t i o n s of
is no d i f f e r e n c e
3~4 go one
a doctrinemap
situation
doctrinemap
(3.9)
that
in o r d e r
further.
morphisms
of K";
Now write
K # for the
D ~ If,f]
of D on A and
between
D ~ (f,f>
Finally, step
m {A,A},
for b.
Similarly in K' a m o u n t s
m
< g,g' )
y < f,g' > (y,l)
Once a g a i n when y' = ¥ we get a d o c t r i n e [y,Y~, and ~0,~i are then doctrinemaps.
It is clear that a d o c t r i n e  m a p D ~ [Y,Y~,
an action of D on Y in K #, is just a D  2  c e l l Y: D  m o r p h i s m s F = (f,~):
A ~ B and G = (g,g):
or equally
F ~ G between the
A ~ B, as defined
in
(3.19) of [13]. 3.5
There is an analogue of (3.7) in which I is an isomorphism:
it can be called either the strong comma object or the pseudo pullback. Each of the main results stated in §i.i has both a "lax" and a "pseudo" case.
In order to make our treatment of these cases formally identical,
we agree to use (f,f) in the strong sense, without i n t r o d u c i n g a new name for it, when h a n d l i n g the "pseudo" case. invertible,
etc.
T h e n the ~ of (3.9) is
3O4
4.
4.1
We first define
between
two d o c t r i n e s
The most E
elegant
then to observe
ff':
a lax d o c t r i n e
(D,m,j)
and
way of doing
map H = (h,h,h°):
(D*,m
it would
,J ) on the be to take
and to form E" from it as we formed K"
= [K,K],
product
Lax a l ~ e b r as
that E" was a strict m o n o i d a l
of the objects
DD' ~ EE';
f:
D ~ E and f':
and finally
to define
D ~
D*
2category
K.
the
2category
from K in §3.2;
category,
D' ~ E'
the t e n s o r
being
a lax d o c t r i n e
map as an
,
object
h:
D ~
D
tensor product.
of E" with the However
we
structure
say the
of a m o n o i d
same thing
for this
in e l e m e n t a r y
A lax d o c t r i n e map, then, consists of a 2natural , ation h: D ~ D t o g e t h e r with m o d i f i c a t i o n s h°,h as in
terms.
transform
m
(4 .i)
i
hh
~ D D
,
h ,
....................
~D
m
satisfying
the axioms Dj
(4.2)
D
m ~ D
•  DD ....
D
~D
Dh
Dh ° D
=
h
~DD
h
h
Dj hD
D
JD Dj
D
. D m
~D
D i
,
305 jD (4.3)
D
DD
h
~D
=
D
tD
Dh f,
JD D
• DD
h
h
> hOD *
hD
D
~DD
,
j D
Dh
,
mD =
~D
Dh
~ DD
DDh I
*
DD
~D
**
D
~
2naturality
a 2cell axioms
maps
which
with respect
by v e r t i c a l
Thus
becomes
and the
*
*
the axioms
a 2category
p:
same
2cells,
and
sub2category
but whose
are the ordinary
Doat
1cells
are
(strict) of these,
as
in [13]§3.6. There
and Doa£. map
from
the obvious
are the m o d i f i c a t i o n s
defined
easily
of doctrines
doctrine
2cells
.
Lax Doct when we define
h ~ k satisfying It has the
~D
*
m
follows
these
and the
*
D
pasting of d i a g r a m s
those with h and h ° i d e n t i t i e s ; maps,
~D
**
we get a c a t e g o r y
to h°,h,k°,k.
same objects
*
D
mD
compose
to be a m o d i f i c a t i o n
with the
*
D DD
satisfies
of h etc.
maps,
~ D
~
,
hD
m
the c o m p o s i t e
lax d o c t r i n e
f~
,
~ DD
•
Dm
Lax doctrine
mD
hhD •
D DD
the
~Dh
DDD
h
hD
hD D
that
=D
>
DD D  Dm
,
m
DDD
,
4.1);
mD
i
~DD
DDD
Dhh 1 ,
D
m
Dm
4.4)
~D
m
are lots
We call
if h and h ° are
of 2categories
the lax doctrine isomorphisms;
intermediate
between
map H = (h,h,h °) a pseudo
we call
it normal
Lax Doct doctrine
or n o r m a l i z e d
if
306
is an identity.
Our arguments below are expressed in terms of
the r e l a t i o n between Lax Doct and Doct, u n c h a n g e d if we replace 4.__22
but they go over a b s o l u t e l y
"lax" by "pseudo" or if we require normality.
For A E K, we define a lax action of the doctrine D on A to
be a lax doctrine map K: defined in §3.2. (3.1) to a:
If k:
D ~ {A,A} where
{A,A} is the doctrine
D ~ (A,A} corresponds under the i s o m o r p h i s m
DA ~ A, then the kind of argument familiar in the
context of closed c a t e g o r i e s s h o w s that k°,k
correspond r e s p e c t i v e l y
to a , a in
jA (4.5)
A
n~
~ DA
D2A
,
DA
~
> Da
A ,
DA
D,
A
a (In detail,
we go from (4.1) to (4.5) by m u l t i p l y i n g
(4.1) on the
right by A and composing on the tail end with the e v a l u a t i o n e:
{A,A}A ~ A.
We go in the other d i r e c t i o n by a p p l y i n g {A,} to
(4.5) and c o m p o s i n g on the front with the unit E ~ {A,EA} where E = i or D2.) (4.2)(4.4)
Equally simple calculations
are equivalent to the axioms DJ A
(4.6)
DA
show that the axioms
(4.6)(4.8) below:
mA m D2A
DA
m DA
.............
~
A

identity,
307
JDA
DA
(4.7)
,, ~.D2A
= identity,
~=~~_ DA =
a
Da JA
~DA
a
*~~A DmA
(4.8)
mDA
mA
D3A
D2A
D2a
.............
J'DA Da
D3A
,, DA
oa
D2A
a ,~, D 2 A ~
a
~A a Da
~ g
a
a
We more commonly use the term lax action of D on A not for the lax doctrine map K = (k,k,k°): triple
D ~ {A,A} but for the above
(a,a,a) satisfying the axioms
(4.6)(4.8).
This latter
d e f i n i t i o n can be used evenwhen A is not an object of K but a m o r p h l s m L ~ K, in which case the first d e f i n i t i o n fails because the right Kan e x t e n s i o n {A,A} of A along itself may not exist; and we do use it in this extended sense. called a lax Dal~ebra;
The A with such a lax action of D is
note that this d e f i n i t i o n agrees precisely
with that of Street in [24]
§2 above.
M o r e o v e r a, for instance,
is
an i s o m o r p h i s m or an i d e n ~ t y precisely when k is ; so we carry over the words algebras.
"pseudo",
"normal" from the lax d o c t r i n e maps to the lax
Finally the strict doctrine maps c o r r e s p o n d of course to
the honest Dalgebras, where both a and a are identities.
308
4J3
For f:
doctrine map P: §3.3.
A ~ B in K, and a doctrine D on K, D ~ (f,f)
where (f,f)
consider a lax
is the doctrine defined in
Because the passage from (4.1)(4.4)
to (4.5)(4.8)
formal, d e p e n d i n g only on the action of E on K s a t i s f y i n g
is purely (3.1), we
can repeat it all at the level of the action of E on K" satisfying (3.5).
We conclude that P corresponds to a lax action of D on f
in K", given by precisely the data (4.5) and the axioms
(4.6)(4.8),
but with A replaced by f and with the 1cell a and the 2cells a,a of K r e p l a c e d by 1cells and 2cells of K" Now consider what this means. r e p l a c i n g a:
(4.9)
f in K"
DA ~ A in K be (a,b,~) as in
DA
Df
Let the m o r p h i s m Df ~
~ A
~
I
'
DE
.~ B .
^
_
Let the 2cells in K" replacing the 2cells a,a of K be the r e s p e c t i v e
pairs
(a,~),
(a,~).
By t h e
definition
o f what a 2  c e l l
is
in K",
these
have to satisfy
jA
(4 .io)
A
jA ~ DA~
A
B
• DA
jB
~ DB
f
1
3O9
mA (4.1i)
D2A
D2A
~ DA
~a
D2f
A
f
~
~
)"
DB
f
~B

B.
b
When it comes to the axioms
(4.6)(4.8), we observe that these are
2cells in K", w h i c h are only pairs of 2cells
in K  s a t i s f y i n g conditions once given
Df
D2B
f
? b
essentially about
~ DA
a
D2f
D2
mA
indeed, but these latter automatic
(4.10) and (4.11).
Thus
(4.6)(4.8)
for the lax action
of D on f reduce to the original (4.6)(4.8) for a,a and for b,b. Putting all this together, we see that a lax action of D on f in K" is the same thing as a lax action
(a,a,a) of D on A (corresponding of
course to the lax doctrine map
D .....(f,f) P a c o r r e s p o n d i n g lax action satisfying
(4.10) and
 {A,A}), ~0
(b,b,b) of D on B, and an f as in (4.9)
(4.11).
We call the pair F = (f,f) a m o r p h i s m of lax Dalgebras F:
A ~ B; it is just what Street calls a "lax h o m o m o r p h i s m "
in
i
f241
§2 above.
We call it a strong m o r p h i s m if f is an isomorphism,
and a strict one if f is an identity.
Of course an argument p r e c i s e l y
similar to that above identifies a strict m o r p h i s m f:
A ~ B of lax
Dalgebras as a lax a c t i o n of D on f in K', and hence as a lax doctrine map Q:
D ~ [f,f].
This shows that,
for lax doctrine maps
just as for strict ones, there is no difference between one into [f,f, and one into (f,f) that happens to factorize through ~:
E f,f ~ ; which could a l t e r n a t i v e l y have been seen by g o i n g
310
back to the definitions of [f,f] comma object.
and (f,f} as a p u l l b a c k and a
This point needs to be kept in mind for our argument
below. (Note that we get the stron~morphlsmsby using the different (f,f) of §3.5.) Finally,
just as we agreed to extend the d e f i n i t i o n of lax
algebra given by (4.5)(4.8)
to the case where A is not an object of
K but a 2functor L ~ K, so we extend the d e f i n i t i o n of m o r p h i s m of lax algebras to this case, by (4.9)(4.11), where f is now 2natural and ~ is a modification. 4.4
Lastly in this hierarchy,
map R: ~y,y~
D ~ ~y,y~, where y:
we have to consider a lax doctrine
f ~ g:
A ~ B in K as in (3.11), and
is the doctrine defined in §3.4.
A r g u i n g as above, we see that
this is the same thing as a lax action of D on y in K #. analogue of the a:
DA ~ A of §4.2 be
(a,b,~,g):
Let the
Dy ~ y, so that as
in (3.12) we have a (4.12)
DA
..... ~
A
=
DA
A v~
Df
Dg
~,
DB
g
...... r
Df
B
f
DB
Y
~ B b
b
Let the analogues of the 2cells a,~ of §4.2 bethe 2cells (a,b) of K #. The axioms
Then ~ and a are as in (4.5), and
(3.14) and
(a,b) and
(b,b) similarly.
(3.15) that these must satisfy to be 2cells
in K # reduce to (4.10) and
(4.11), with their analogues for g,g.
Finally the laxaction axioms to (4.6)(4.8)
g
(4.6)(4.8)
for a,~ and the analogues
for (a,b) and (a,b) reduce for b,b.
Summing up, we
have lax D  a l g e b r a structures on A and B, m o r p h l s m s F,G:
A ~ B of
these where F : (f,~) and G = (g,g), and finally a 2cell y: satisfying
(4.12)
f ~ g
(which is identical with the (3.19) of [13], d e f i n i n g
D2cells in the case of honest Dalgebras).
311
We therefore even in this where L ~ K,
call
lax case;
such a y s a t i s f y i n g
and we extend
the lax D  a l g e b r a s
Now that we have we make
the various
definition
A and B are not objects
in which case ~ is of course
D2cells,
this
(4.12)
a D2cell to the case
of K but
2functors
a modification.
lax Dalgebras,
morphisms
of these,
and
LaxDAlg of these elements, d e f i n i n g
a 2category
c o m p o s i t i o n s by the evident
pasting
operations;
restrict
to the strict m o r p h i s m s
categ o r y
LaxDAlg,. There is a similar r e s t r i c t i o n to the strongmorphisms.
4.5
It is easy to see,
inclusion
and c o t e n s o r object
although
Doat in Lax Doct
of
products.
of lax Dalgebras,
if we
tedious
to write out,
is 2continuous~
It follows
that
we get the
limits
D, r e g a r d e d
Lax Doct, admits a 2  r e f l e x i o n H = (h,h,h°):
of
that the
it p r e s e r v e s
a doctrine
sub2
as an
D ~ D
into
Doct, p r o v i d e d that the a p p r o p r i a t e solutionset condition is satisfied
for the given
a lax d o c t r i n e
D.
map H:
That
D ~ D
is to say,
such that
there
is a d o c t r i n e
any lax doctrine
map
D
K:
and D ~ E
,
is of the form K : tH for a unique We get the
same
2continuity
we require
normality;
I do not which,
for a given
doctrine
if we replace
and exactly
intend
strict
to study
the
"lax" by
map t:
"pseudo",
same c o n s i d e r a t i o n s
in this
D, the s o l u t i o n  s e t
paper
condition
D
~ E. or if
apply.
the conditions
under
is satisfied
and the
,
reflexion
D
exists.
Certainly
Cat/~ it is immediate that D the m e t h o d s
of Barr Ill
say K is locally question
too because
exists,
or of Dubuc
presentable
aside b e c a u s e
when D is Do
for a small
as we see b e l o w in §4.10.
[5]
will
show D
and D has a rank.
I haven't
thought
I suspect that it doesn't
club D in
to exist
I suppose
if
In part I leave
out the details;
really matter:
but
it seems
the
in part to me
,
likely
that D
Cat, or that D universe
always exists
exists
as a d o c t r i n e
on a suitable
when D is g i v e n on K.
of e x t e n s i o n s
of D a d u m b r a t e d
on CAT when D is one on
completion
The q u e s ti o n in the
second
of K in a b i g g e r
is c o n n e c t e d paragraph
with that
of §3.2,
and
312
really Our
deserves
concern
a fuller
in this
treatment
paper
is w i t h
in its o w n r i g h t the
consequences
at a n o t h e r of the
time.
existence
,
of D , a n d
the
club
case
is
sufficient
to s h o w that
our c o n s i d e r a t i o n s
a r e not v a c u o u s . A final the p r o o f
point
of T h e o r e m
exist.
If it d o e s
affects
its u t i l i t y
not,
there
by
merely
be d e n i e d
should
have
which
would
purely
ly of the
"size"
4.6
"normalized the u s e
theorem
called
universe,
that
theorem.
And
remains
true, and
of [i0]
above,
which
noone clearly
to p s e u d o  D  t e r m s ,
would
doubt
we
should
stand,
and
case, that
such
such when
be v a l i d
the m e t a t h e o r e m
does
replacing
§i as t h e y
remain
D')
if it d o e s
so on;
would
for
in no w a y
in the p s e u d o  a l g e b r a
But
though
( there
Dcategory"
of the r e s u l t s
nuisance.
D
in [I0]
independent
involved
is felt
stated. b a c k to the m a i n we h a v e
is g i v e n
in b i J e c t i o n
Thus
of that
pseudo
of D  e v e n
A in K  s o m e t h i n g
are
that
bigger
them over again
D'terms
Coming
an algebra
that
dlagramarguments,
from
than
in some
that
I assumed
paper,
for the p r o o f
be a n a w f u l
translated
rather
so o n l y
to p r o v e
formal
vein:
3.1 of that
is no d o u b t
" D~category"
in t h i s
the D  a l g e b r a s
as b e f o r e
to t h e h o n e s t
action
a = a
*
Otherwise
put,
every
D action
a
.hA,
a
just
to the :
*
~
.hA,
lax D  a c t i o n
identify
map
last t:
the
the D  a l g e b r a s
paragraph.
D
lax d o c t r i n e
~
maps
{A,A}; K:
lax D  a l g e b r a s .
lax a c t i o n
D A ~ A,
a = a
we
in the
doctrine
K = tH w i t h
in K are
K corresponds
(4.13)
anticipated
by a strict
under
point,
(a,a,a),
the r e l a t i o n
a = a
*
D ~
Such and
{A,A}.
Moreover
while
these
if
t corresponds
K = tH t r a n s l a t e s
into
0
.h A.
is of the
form
(4.13)
for a u n i q u e
.
Similarly
a morphism
F*
(f,f):
A ~ B of D algebras
is
,
given map
by a s t r i c t
D ~ (f,f),
lax D  a l g e b r a s .
doctrine
which The
map
D
corresponds connexion
~ (f,f)
and hence
to a m o r p h i s m
between
F and F
b y a lax d o c t r i n e
F : (f,~): is g i v e n
A ~ B of by
313
a (4.14)
DA
=
A
I
Df
DB
hA
DA
DB
B
factorizes
~ = i
~
If,f]
if and only
if and only
is, F is strict Finally
maps
If,f]
through
(4.15)
that
e:
D ~ [y,y~,
(4.16)
~
trivial
is that
produces we have
also
the
...... b
F
of v e r t i c a l diagram the
if F
map,
does
.
~
so; w h e n c e
correspond
to lax d o c t r i n e
that
~ G
are just the D  2  c e l l s
(4.14)
the only
pasting
case
and
that
of diagrams
(4.15)
D * Alg [resp.
LaxDAlg
2functor
[resp.
y:
F ~ G.
respect
the
is not a b s o l u t e l y (4.14),
which
of the same kind by the n a t u r a l l t y
2category
D
F
is so.
~ ~y,y~
2category
know the forgetful
for a unique
doctrine
if D ~ < f , f >
the c o r r e s p o n d e n c e s
identified
essentially
y:
of composition;
another
D B
A
~* = I,
doctrine mapsD
Moreover kinds
if
and we conclude
(4.14)
is a strict
if and only
the D 2cells
various
a
hB
we can say that any F is of the form
Since m o r e o v e r
* D A
D f
Df
b Again,
~
clearly
of h.
Thus
D * A l g,], namely
LaxDAlg,l.
to K, we do indeed
have
as
Since we an implicit
,
d e t e r m i n a t i o n of D . (This whole section remains v a l i d for the stronger of §3.5.) 4.7 For the purposes of our proof, we need the results of §4.6 not only
for a l g e b r a s
2functors
A:
A that
L ~ K (L being
for such an A the right not exist, argue
we cannot
objectwise,
are o b j e c t s K itself
Kan e x t e n s i o n
argue d i r e c t l y
using
the r e s u l t s
of K but
for a l g e b r a s
in our a p p l i c a t i o n s ) . {A,A}
of A a l o n g
as in §4.6. of §4.6.
itself
Instead
that are Since need
we have
to
314
Let A:
L ~ K then
(a,a,$).
For e a c h
and h e n c e
as in
X E L we get a lax D  a c t i o n
(4.13)
a unique
*
aX = a X.hAX, is that
a
To Because
aX = a X.hAX,
the
unique show
D action
with
lax
action
(aX,~X,aX)
a X on AX such
on AX, that
~
is 2  n a t u r a l ,
of c o u r s e
be a lax D  a l g e b r a ,
one
and
aX = a * X . h ° A X .
so that
a
satisfying
the n a t u r a l i t y
a is n a t u r a l ,
is i n d e e d
we have
What
has
to p r o v e d
a D action
on A, and
(4.13).
of a , c o n s i d e r
commutativity
@:
X ~ Y in L.
in
aX (4.17)
DAX

~AX
, ,
A¢ DA$ I
DAY
~ AY
;
aY and A S is in fact
a strict
(4.10)
being
Hence
and by
(4.11)
(4.14)
and
morphism
satisfied
(4.15)
of lax D  a l g e b r a s ,
because
we have
a and
the
axioms
a are m o d i f i c a t i o n s .
cormmutativity
in
a X (4.18)
D AX
D
AX
A¢
A~
DAY
, aY
~
AY
,
,
showing
the n a t u r a l i t y
X ~ Y in L. whence
Then
of a
As:
A S ~ A@
it is a D  2  c e l l Next,
of these.
Dalgebras
in K, w h i c h
morphism
by
let A,B:
a morphism
.
its
2naturality,
is a D  2  c e l l
(4.16),
giving
by the
the
FX = (fX,fX):
therefore
¢ ~ ~:
2naturality of a
and F = (f,~):
AX ~ BX is a m o r p h i s m
corresponds
of D *  a l g e b r a s
let ~:
2naturallty
L ~ K be lax D  a l g e b r a s
Then
F * X = (fX,~ *X)
For
as
in
in K, w h e r e
(4.14)
of a, . A ~ B of lax
to a u n i q u e
~X = ~ * X . h A X .
If
315
F is strict,
so is F
, and t h e r e
general
we m u s t
s h o w that
case
is no m o r e
f
to prove;
is a m o d i f i c a t i o n .
but
in the
This m e a n s
the
equality
,
a X .................
D AX
D fX
a X AX
~
t " 1 f X
D BX
~
, b
we h a v e
because
~
this
because
the
correspondence
correpondence
, b Y
(4.14)
that
~ BY
respects
f X corresponds
~ is a m o d i f i c a t i o n
fY
f Y
D BY
BY
At the next level, nothing
D fY
Y
because
under
(4.17),and
....
~ AY
DAY
Be
D BY
which
a Y
BX
D Be
AX
D A¢
fX
,
~
D AX
;
composition,
to ~X and
(4.18)
to
by h y p o t h e s i s .
of D  2  c e l l s ,
there
is by
(4.16)
to prove. We
conclude
that
the
results
of
§4.6,
expressed
in the
form
^
that
every
D action
lax D  a c t i o n a
, etc.,
present, w i d e r , 4.8
We
(a,~,a)
remain
true
is of the when
form
we take
(4.13)
for a u n i q u e
the a l g e b r a s
in the
sense.
can n o w p r o c e e d
rapidly
to the p r o o f
of our
first
main
result. Theorem
4.1
Suppose
that~
Let
considered
H = (h,h,h°): doctrine (4.19)
map sH
D be a d o c t r i n e
D ~ D*
=
1,
as an o b j e c t
into
satisfyin~
on the
Doat.
Let
of s:
2complete
2cate$ory
Lax Doct, D a d m i t s
K.
a reflexion
D * ~ D be the u n i q u e
strict
316
which
exists
a lax one. and qh "lax"
because Then
= i.
there
The
same
D ~
The
comments
at the
and
the d e t a i l s
from
last
is t r u e
maps,
(4.20)
sh
(4.19)
=
i,
in
to t h i s u n d e r the
The (4 • 13)
The
(4.20)
be w r i t t e n
sh
=
m: m
lax a c t i o n
DD
,~
that
the a x i o m
morphism
to H
(4.4).
s~ = i
proof,
first
in the
from
the
paragraph light
of c o m p o s i t i o n
of
of §4.5,
§3.5.
of lax
as
map,
(4.20),
:
D
D
i.
m.sD:
the
D D ~
D is a D  a c t i o n
on D w h i c h
lax action
~ D
of D
(m* . h D * , m* . ~ ~D*, of
m
= D
DD
corresponds
(m,l,l);
that
is to
(4.1)
may
m
on i t s e l f
corresponds
* . hOD * ) of
be w r i t t e n
D
on
D
by
*.
as
h
~ D
m
H = (h,h)
(4.10)
that
or if we r e p l a c e
is i m m e d i a t e
lax D  a c t i o n
diagram
hD
reduces
The
such
normality~
in the
sh ° =
h
DD
I ~ hs
and afortiori
D 2 ~ D of D o n i t s e l f .
Dh
I assert
i,
is, b y
action
second
those
may
map
is a n i s o m o r p h i s m .
by the d e f i n i t i o n
D action
to the
from
following
of [13] .
(4.13)
strict
case ~
s is a d o c t r i n e
(3.8)
q:
of the t h e o r e m
of the
that,
Because on D, as
sentence
doctrine
if we r e q u i r e
last
end of §4.1,
Observe doctrine
D is a s t r i c t
is a m o d i f i c a t i o n
by "ps___~euJg"; i n t h e
Proof.
say,
i:
is a m o r p h i s m
in t h i s
case reduces
It f o l l o w s
= (h,h):
D ~ D
as
in
D ~ D to
of lax D  a l g e b r a s .
(4.3),
(4.14)
that
of D  a l g e b r a s
and
the a x i o m
there
For (4.11)
is a u n i q u e
satisfying
317
(4.21) m
hD
DD
D
, ~D D
DD
=
sD
m ~D
~DD
2Y Dh
h
h
Dh
D*h
h
@@ DD
,
~DD
other
and hence
a strict
and
the
(4.22)
sH
replaced
The
.
take
The
so that
respects
morphism Since
of D  a l g e b r a s ,
sH = i by
composition,
(4.1).
to
,
2.5;
(4.20)
we h a v e
D since
Since
by
the q:
i ~
of c o u r s e
by D, a n d H
(2.4),
the
s is a d o c t r i n e
sh ° = i by we h a v e
with
B replaced
to be a m o d i f i c a t i o n
2.5 a g a i n
(2.22),
Theorem
2.5 n o w b e c o m e s ,
is j: i ~
2.5 is t h e r e f o r e
by Theorem
to a p p l y
by D i = D
r of T h e o r e m
D, w h i c h
According is the
of lax D  a l g e b r a s . (4.14)
~D
,
m
~ D is a s t r i c t
in a p o s i t i o n
it to be the h ° of
st = i,
s: D
DA r e p l a c e d
: i ~
~ of T h e o r e m
DD
1.
by D ,
sj
hand
correspondence

by H
composite
,~
hD
morphism
W e are n o w D replaced
DD
m
On the
since
D
,
hD
sq
j
(4.20),
map.
~ hj;
a n d we
we h a v e
= i.
H s produced
by T h e o r e m
2.5
composite
(4.23)
DD D *j
,
D
Y
,
~D
h ,
*
D
D
Dh ~
It r e m a i n s
of
only
h we h a v e
D
to g i v e
D*h°.h
a proof
= hD*.Dh °
D
that
so that
q h = i.
(4.23)
But
by the
2naturality
composed with
h becomes
318
D
(4.24)
Dj
hD
DD
* ~ D D
Dh Dj*~
sD
~
D h
, DD
,
~
the
composite
(4.21),
whence
, , DD
h ,
,
~' D
m
o f the
(4.24)
~~ D
_, h
hD
however
m
DD
is the
two rectangles identity
by
o n the r i g h t
(4.2).
This
is h as in
completes
the
proof. 4.9 of
In m y p a p e r §3.1, a n d u s e d
[I0]
in the
above
proof
in t h i s
volume,
of T h e o r e m
3.1,
I stated
only
at the
one v e r y
end
small
part
,
o f the
above
theorem:
namely
that
s:
D
~ D had
a right
inverse
h,
,
as
in
(4.20).
pseudo
case, In
The
D
which
in q u e s t i o n
is the c a s e
~3.3 of II0]
here
that
I called
is t h a t
arises
for the
normalized
in [i0] .
the d o c t r i n e
D flexible
if s:
D
~ D
,
had a right notion
inverse
was u s e d
doctrine
D
is a l w a y s
equally
D ~ D
in T h e o r e m
The p r o o f applies
t:
that
a
3.3 of [i0] .
flexible:
we n o w
is e l e m e n t a r y ,
in t h e
was
lax case
as
not
(strict) It was
stated
justify
map;
there
this
that
the
this.
depending
in the
doctrine
on T h e o r e m
pseudo
case,
4.1,
and
w i t h or w i t h o u t
normality. Proposition
4.2
as an o b j e c t reflexion q:
D
of
Let H:
Lax D o c t ,
D ~ D into
of D , c o n s i d e r e d
~ D
be the u n i q u e
there
is a s t r i c t
Proof
Since
Doct;
and
as a n o b j e c t strict
doctrine
be the r e f l e x i o n
map
p:
let K: of
doctrine D
~ D
D
o f D,
considered
~ D
be t h e
Lax D o c t , map
such
with
into Noct.
that
qK = i.
Let Then
qp = i.
** KH:
D ~ D
is a lax d o c t r i n e
map,
and
since
,
H:
D ~ D
doctrine since
is t h e r e f l e x i o n , map
qK = i.
p:
D
~ D
Since
f o r the r e f l e x i o n
Then qKH
qp a n d
H gives
we h a v e K H = p H for a u n i q u e = qpH;
I are b o t h
qp = i.
D
but
strict,
again
qKH
strict = H = iH
the u n i q u e n e s s
property
319 In the pseudo case,
since q is an e q u i v a l e n c e of endo2
functors by T h e o r e m 4.1, it follows that pq a i. have not proved,
I suspect, but
that the i s o m o r p h i s m pq m 1 can be chosen to be a
doctrinemodification,
so that D
This would be very convenient
and D
are equivalent doctrines.
if it were true, and the q u e s t i o n should
be looked into. 4.10
We c o n s i d e r finally the special but important case w h e n
D = Do
for some club P in Cat/~.
and §i0 of [9]
We again refer the reader to §i
above for general facts about clubs of this kind.
To give a lax D  a l g e b r a structure, w h i c h we shall also call a lax P  a l g e b r a structure, functor a: {A,A}
on a c a t e g o r y A, we have first to give a
PoA ~ A, or equally a m o r p h i s m P ~ {A,A} in Ca£/~; here
is now the "rich e n d o  f u n c t o r category" of [9]
right Kan e x t e n s i o n of (3.2) above. ITI:
So for T C P we have to give
A n ~ A where FT = n; and for f:
g e n e r a l i z e d natural t r a n s f o r m a t i o n
§9, and not the
T ~ S in P we have to give the
IfJ:
ITJ =
ISI of type Ff.
Since
this is to be a functor, we must require
(4.25)
Ifgl = Ifr
Igl and
Iii = i.
Next we have to give a as in (4.5), a natural t r a n s f o r m a t i o n with components
ILl(A), that is, a natural t r a n s f o r m a t i o n
In the normal case we require ~ = I, which involves d e m a n d i n g that It1 = i.
In the pseudo case we have to provide a with an inverse,
we must also give a natural t r a n s f o r m a t i o n
$':
ltJ
and demand that
i,
so
320
a Similarly
'
=
i
and
a'a
=
for the a of (4.5);
aT[SI...Sn[AI...Am]
i.
it is to have components
:
ITI(ISII(AI''') ..... ISnl(...Am))
IT(SI...Sn)I (AI...Am),
with inverses a' provided
in the pseudo
case.
All these data are
then to satisfy the further axioms c o r r e s p o n d i n g It is clear from [9]
of D
such a lax D  a l g e b r a
is itself
in Cat~l, which we have in effect just
an algebra for a club O described
§I0 that
to (4.6)(4.8).
by its generators
and relations.
Explicitly,
the objects
are generated by objects T in bijection with the objects T of
D, and with FT = FT. the normal
These are subjected
to no relations
case, where we impose the relation
for the morphisms
of D
consist
of an f:
[ = i.
except
in
The generators
T ~ S with r~ = rf for each
^
f:
T ~ S in D; of an a:
~ ~ ;, with ra = i, which is to be omitted
in the normal case; and of an
aT[SI...S n]
:
T(SI ..... ~n )
~
T(SI ..... Sn)'
with Fa = i, for each object T[SI,...,S n] of PoD. these are to be augmented by further generators in the reversed these generators
senses.
are fg = fg and ^
relations
The relations
In the pseudo case,
a' and a'T[SI...Sn]
between expanded
i T = I~
corresponding
instances
to (4.25);
the
_ _
aa' = i, a'a = i, aa' = I, a'a = 1 in the pseudo
and finally the relations
of
corresponding
of these for instance asserts
to (4.6)(4.8).
the commutativity
of
case only;
The first
321
l
aT[ 1 . . . II
T D'DD
D'D
identity,
~
~ DD v
m'DD (5.10)
D'D'DD
D'DD
~
~
D'D'DD
~ D'DDD' 
l
D'~
DD v
~
1
~D' Dm '
........
~ D ' D D ' ~
DD'D'
DD'
pDD (5.11)
D'DDD D'Dm ~
...........
~ DD'DD
pD
DD'm~ Dp
D
'DD
...........
,L
~
.... ~ DDD'
DD'D
DVD
~
DD v
P DDp DDD'D .....
D 'DDD ~rD
DDDD '
I mDD '
mD 'D Dp
D 'DD
DDD '
DD'D
> D'D ....................
,ol,
327
We now Justify the above assertion. in (5.1) is by [13] DU',.
By [13]
§3.3 to give an action q:
Proposition
give a 2natural
p:
p:
D'D ~ DD'
and for a D'algebra
extension D, and write DA
In doing so we use A both for an object of K (A,a')
consisting
D'A ~ A.
of the object A together with
We then have
DA = (DA, D a ' . p A ) ,
the D'algebra
with object DA and with D'action
D'DA
~ DD'A pA
~ DA. Da'
For a morphism F' = (f,~'):
A ~ B of D'algebras,
D'morphism
(5.14)
DF'
=
(nf, n['.pA)
as in pA (5.15)
D'DA
Da ' ....... ~ DA
~ DD'A
)D'Df I D'DB
DD'f 1
pB
for
form of D, in terms of p; but we may as
well give at the same time its canonical rather than D,A, etc.
The conditions
of
at once into (5.3) and (5.4).
We give the explicit
(5.13)
q as above is to
D'D ~ D'U',F', where F', is the left adjoint
q to be an action translate
(5.12)
D'DU', ~ DU', of D' on
2.1 to give a 2natural
U',; that is, to give a 2natural
the D'action a':
First, to give D, as
~ DD'B
Df
D~'
Db '
DB
we have a
328
Finally a D'2cell y:
F' = G' gives a D'2cell Dy:
DF' =~ DG'
directly by
(5.16)
~
~.
=
The 2functor D, is what we get by r e s t r i c t i n g D to strict D'morphisms;observe
that if f:
A ~ B is a strict D '  m o r p h i s m then
D,f = Df is just the strict D '  m o r p h i s m Df:
DA ~ DB.
Since what we
have said about D, is all justified by our general remarks above, all that needs separate v e r i f i c a t i o n is that when F' is not strict, that are not strict,
(5.15) really is a D '  m o r p h i s m
(5.16) really is a D'2cell w h e n F',G'
and that D really is a 2functor:
The next thing we w a n t e d was 7,: The last requirement jA:
all this is easy.
I ~ D, with U',J, = jU',.
forces the component j,A:
A ~ D,A to be
A ~ DA; the desired J, exists p r e c i s e l y when JA:
indeed a strict D'morphism, Once we have this,
and this c o n d i t i o n reduces to (5.5).
it is immediate that JA:
A ~ DA is 2natural not
only for strict D '  m o r p h i s m s A ~ B but for all; 2natural j:
(5.17)
~A
A ~ DA is
so we in fact get a
i ~ D with the same components:
=
],A
= jA;
of course we have U'j = jU' Then we wanted m: requirement to be mA.
(5.18)
~2 ~ $ w i t h U'm = mU'.
forces the m o r p h i s m  p a r t
The last
of the component mA:
D2A ~ DA
Thus
mA
=
(mA, m'A)
for some invertible
2cell m'A
(since mA is to be strong).
of m'A is d e t e r m i n e d by the r e q u i r e m e n t
that m be 2natural.
The form Indeed,
329
mere f:
naturality A ~ B,
in A of mA,
suffices
to be a':
and
to fix the
D'A ~ A, we e a s i l y
that m e r e l y f o r m of m'A.
see that
pDA
(5.19)
D'DDA
,,
for
m'A
strict
First,
taking
is of the
form
DpA
,
~ D D ' D A
DDD'A
~
~A
mA
~ DD'A
~ DA
pA
we e a s i l y more
~A;
see that
is n e e d e d The
w must
to m a k e
further
for a D '  m o r p h i s m (D,m,j) and
satisfy
(5.11)
pseudo
in e l e m e n t a r y
the a x i o m s
look
terms
with
= (a,a'):
D'a
.......
are that mA
(5.7)
and
classical
case
satisfy
reduce
w = i, but
w.
2category
involving
D,D',p,w
A is a D '  a l g e b r a a:
DA ~ A. in v i e w
of
DAlg and A
elements
(5.9),
analysis
its
of a
elements
of K. a':
D'A ~ A)
is to be a D '  m o r p h i s m
(5.13)
the
has
~ DA
a'
a
" 1 ~ A
a t
(5.8),
not here).
D'action
Da '
l
the a x i o m s
The l a t t e r
~DD'A
D'A
to
and d e s c r i b e
(with
no
 and that
in terms
of p and
Then
is simple.
(5.10)
these
D ' A ~ D'B,
(5.6).
of the a b o v e
at the
a Daction
in
our j u s t i f i c a t i o n
DA ~ A, w h i c h
D'DA
to
as
the v e r i f i c a t i o n
for a d o c t r i n e ;
pA (5.20)
A ~ B to be D'f:
we do n e e d
in the
law
A Dalgebra together
m 2natural:
completes
We n o w
f:
be a m o d i f i c a t i o n ,
these reduce
distributive
5.~
Da'
taking
things
(automatic
This
next,
A ~ B
~ DDA
mD'A
D'DA
for an i n v e r t i b l e
f:
DDa'
......
D'mA
D'morphisms
form
330
There
are
really
four axioms
to be s a t i s f i e d :
is a D '  m o r p h i s m ,
and
the
second
the
first
two
two
say t h a t
say t h a t
a
it is a D  a c t l o n :
j 'DA (5.21)
DA
~ D'DA
~ DD'A
DA
D'A
=
identity,
"~A
m'DA (5.22)
D'D'DA
,
~
D'DA
~ DD'A
D'A
D'D'DA
~ D'DD'A
............
~ D'DA
~
~ DD'A
D'~' D'D'A

D'A
~ DA
~ A ,
T
D'A
=
D'jA Da '
D'DA
A
~'
~
a
(5.23)
~DA
DD'A
g') ]
~ D A
...............
D'a
a
D'A
..............................
a'
~A
identity,
331
(5.24)
pDA ....... DD'DA
D'DDA
DDa '
DpA DDD 'A
~ DDA
> D'mA
~A
D'DA
......~ DD'A
~ DA Da '
pA
a
D'a
D'A
~A a'
pDA
DpA
D'DDA
DDa'
DD'DA
DDA
DDD'A
DD'a[
D'Dla D'DA
D~'
DD'A
~
pA
f
DA
Da'
> a
D'a[
a'
D'A
Of these,
(5.21) and
a.jA = i using using
(5.20),
D'action.
(5.22) are straightforward;
(5.20) and (5.17 (5.19),
the equalities
and (5.15).
(5.21),
expresses
a.mA = a.Da,
since a' was given as a
level already gives new information; that a:
1cell equality
in (5.23) and
see by composing
expresses
Note that at the level of 1cells
(5.22) are automatic,
the information
equalities,
; (5.24)
(5.23)
The same is not true of (5.23) and (5.24),
at the 1cell
convenient
A
DA ~ A is a Daction. (5.24);
(5.23) with J'A and
where equality
in fact precisely
(Clearly this implies
and it is implied by them, as we (5.24) with j'D2A.)
It is very
to separate the 1cell from the 2cell information for example
in the kind of reasonings
in such
used in §4.3 above.
332
Thus
in fine to give a D  a l g e b r a
A of K, w i t h a D  a c t i o n a' as in (5.20)
A and a D '  a c t i o n
satisfying
write the D  a l g e b r a
A as
the four axioms
(A,a,a',~').
we d e s c r i b e d such an algebra, both a D  a c t i o n structure
of a D'morphism. Now we consider
It is a pair F : (F',~) D'2cell
and to give a 2cell
(5.21)(5.24).
terms,
the D  a c t i o n
being given
with
the
this precise.)
it is to give a D  m o r p h i s m
where F'
We may
as one p r o v i d e d
We have now made
what
is a D '  m o r p h i s m
F:
A ~ B.
A ~ B and f is a
as in
~
(5.25)
a
DA
~A
DF'
f
~B If F'
a',
(In §i.I of the I n t r o d u c t i o n
in rough
and a D'action,
A is to give the object
is itself
the pair
(f,~'
F' ~B
:
A ~ B then as a 2cell
~ is of the
form
a
(5.26)
DA
~A
Df
f DB
f ~
B.
b
The r e q u i r e m e n t
that
f actually
be a D'2cell
is
(cf. [13] (3.19))
333
Da '
pA D'DA
(5.27)
.....
DA
~ DD'A
_> Df' D'DB
DB
~ DD'B
D,b~~B
f
A
Db' b'
D'B
,.b B
b'
pA =
D'DA
D'DB
Da ' . . . . . .
~ DD'A
D'f
~ DA
D'A
~ A
D'f
f,
D'B
~B b'
Finally they
the a x i o m s
assert
axioms
only
F:
= (f,f,f')
s u c h that
we
y:
the
condition
y:
~
in fact
level
(B,b,b',b')
of
what
for
(g,g,g')
is,
2cells,
Thus
reduce
since to the
in f i n e to g i v e
is to g i v e
and
(5.27)
a D2cell
since
y:
it is a p u r e
a 2cell
F ~ G and a D'2cell
y:
F' ~ G';
it to be a D  2  c e l l is just
a Dmorphism,
a
a triple
A ~ B is a D  m o r p h i s m ,
of all to be a D '  2  c e l l
(f,f,f')
D2cell
~
F = (f,~):
consider
for it to b e a D  2  c e l l just
at t h e
A ~ B is a D '  m o r p h i s m ,
Lastly first
to be
to be a D  m o r p h i s m .
(A,a,a',a')
= (f,f'):
has
an e q u a l i t y
for F = (f,~)
Dmorphism
F'
for F = (F',f)
y:
y:
y:
is s a t i s f i e d . F ~ G: but
A ~ B is.
then the
equality
It
condition
of 2  c e l l s ,
F ~ G. So a D  2  c e l l f ~ g that
F' ~ G'
is at o n c e a
334
~.4
We intend
doctrine
on K.
so that vertex a:
to exhibit
The
it becomes the {A,A}
first
D ~ {A,A}
of §3.2;
we get
If we compare
§§4.24.4,
while
will
with
the present
now suppose K to have This
together
small homs
k:
D ~ D
doctrine
{A,A}
maps
corresponding
similarly.
§5.3 above
However
to a'
we abstract
by an a r b i t r a r y is analogons
to §4.1.
to
Of course we
and to be 2complete.
leads us to the f o l l o w i n g
of d o c t r i n e  m a p s
not the actions
something
replacing
§4, then
(5.20)
vertex D'D and with t e r m i n a l
§5.4 is analogons
, j ) on K, we define
modification
with
for a
is to t r a n s f o r m
corresponding
transform
in this way,
doctrine.
this
with leading
D'A ~ A but the
(5.21)(5.24)
the t r a n s f o r m
in doing
as the algebras
it will then involve
and D' ~ {A,A},
The axioms
(D , m
step
a diagram
DA ~ A and a':
Dalgebras
notion.
a map K from and k':
For any d o c t r i n e
(D,D',p,w)
t_2o D
D' ~ D , t o g e t h e r
with
to consist a
k as in
(5.28)
D'D k'k
P ~
D D
~ DD' kk'
D~D
D, satisfying 1cell
the following
level):
four axioms
(which are all automatic
at the
U
U
S . z °
U
~J
ct lJ.
j,J. C~
t~ t~
~3
t~
tZJ
jJ°
U
W~ t~
~3
~t
C'F
C~ (1)
U U
i
U
U
t',J v
'J1 L~J 0 v
f,~ 01
336
pD'
D'p
(5.31)
D' ~'D
DD'D'
D'DD'
I
m'D
Dm '
P
~ DD v
D D
I
kk'
k'k ^
k
D D
D D
D'p
pD' ....~... D'DD'
D'D'D
D'k'k I
£ D'k
kk
D'D D
D'D D
kD '
^ ~ kD'
D D D'
'm* D'D
~ DD'D'
D*k D D D
DD
m\ /" D
D D D' ' D D'
DD
kk' D '
D'
337
Dp
pD D'DD
(5.32)
DDD '
'~ DD'D ~>
D'm
imD'
:~' DD'
~'D >
k'k
kk'
^
k
DD
D D
Dp ..
pD : DD'D
D 'DD
k
'kD1
....;D ... >
DDD
/ k D, ~ D D D
~ DDD'
> ^ Dk
'k
DD D
DD D D k
,
mk D D,
D ,D ,D ,
DD
~,~DD ~
DD
D
Dkk'
338
If K = (k,k',k) and H : (h,h',h) D
,
we
define
a
2cell
H ~
K to
consist
of
are maps from (D,D',p,~)
to
doctrinemodifications
0:
h ~ k and p': h' ~ k', satisfying the evident axioms with respect ^ ~ ~ to h and k. Thus we get a category Map((D,D',p,~),D ),or M(D ) for short.
A doctrine map t: M(t):
M(D*) ~ M(E),
D
~ E induces a functor
sending (k,k',k)
to (tk,tk',tk).
A doctrine
m o d i f i c a t i o n ~: t ~ t I induces a natural t r a n s f o r m a t i o n M(t) ~ M(t I) ^ whose (k,k',k)component is (Tk,Tk'). Thus M constitutes a 2functor Doct
~
CAT.
Now it is easy to see that a map K: corresponds
exactly to a Dalgebra
(5.20)(5.24);
(D,D',p,~) ~ {A,A}
structure on A as described in
we have only to replace D
by {A,A} in (5.28),
m u l t i p l y on the right by A, and compose with the e v a l u t a t i o n e:
{A,A}A ~ A, to get the s i t u a t i o n of (5.20).
(5.30),
(5.31),
respectively;
(5.32) easily reduce to (5.21),
The axioms (5.23),
(5.29),
(5.22),
(5.24)
of course they were set up to do just this.
But now a map from (D,D',p,~)
to the doctrine { f,f} of §3.3
gives the analogue of (5.20) with f r e p l a c i n g A a n d the diagram now living in K".
So a,a' get r e p l a c e d by actions of D,D'
that is to say, by enrichments of f to a D  m o r p h i s m D'morphism (f,~'), and a' gets r e p l a c e d by a 2cell
on f in K",
(f,~) and to a (a',b')
in K";
the c o n d i t i o n for this to be a 2cell in K" is precisely
(5.27).
axioms
separately.
(5.21)(5.24)
Hence to give a map
reduce of course to those for a',~'
(D,D',p,~) ~ (f,f) is to give D  a l g e b r a structures
to A and to B and to enrich f to a D  m o r p h i s m Finally,
The
for T:
f ~ g:
(f,f,f'):
A ~ B.
A ~ B, a map from ( D , D ' , ~
to the
doctrine ~y,T~ of §3.4 is equally easily seen to be just what makes y a D2cell.
339
Note that a map from §3.3,
or equally
[f,f],
a map to ( f , f }
corresponds
identities. clearly
(D,D',p,w)
This
that happens
to a D  m o r p h i s m
in which both
We now pass on to the a n a l o g u e s
that the
2functor M:
is likely
Doct ~ CAT
particular
equally
for symmetric
of §§4.54.7.
which
It is clear
it is t h e r e f o r e condition
the conditions
here.
in §6, where D comes
club
Dmorphism,
solutionset
in §4.5 c o n c e r n i n g
to be so apply
applications
through
f and ~' are
is 2continuous;
if the a p p r o p r i a t e
The remarks
of
f' arbitrary.
5.5
fied.
[ f,f]
to factorize
is not the same as a strict
has ~ = i but
2representable
to the d o c t r i n e
under w h i c h this
It is certainly in C a t / ~
from a club
monoidal
categories.
is satis
so in our and D' from the
We h e n c e f o r t h ,
suppose
it to be so, and we h e n c e f o r t h
the r e p r e s e n t i n g unique
doctrine
map
We want case we have because
map.
Thus
t:
D
we have
be an isomorphism, isomorphism. difference
two cases
be stron$,
D'A/g**
in the former;
This
corresponds
and hence
simultaneously.
the lax case.
There
with
D restricts
to the to
to be an
is no formal
M is 2continuous
them t o g e t h e r
However,
a' in (5.20)
k in (5.28)
case.
The
where we retain only
to r e q u i r i n g
to r e q u i r i n g
the two cases;
we treat
to be
~ E is tK for a
the d o c t r i n e
of D'A/g
We call this the Pseudo
between
~ D
~ E.
supposedmAto
D'morphisms.
(D,D',p,~)
(D,D',p,w)
so far may be called
one on the s u b  2  c a t e g o r y strong
any map H:
in fact to consider
considered
take K:
in the latter as
identical
notation,
,
although
of course
it is a d i f f e r e n t
D
in the latter
case.
However,
,
in our prime club,
application,
as it does
will not in the pseudo
in the lax case:
direct
proof
course
in the pseudo
and all our
D
of its existence
2cells
(cf.
and
case come from a
I have therefore
not the
§1.4 of the Introduction).
case we ~ive (f,f) are isomorphisms.
its
stronger m e a n i n g
same Of
of
§3.5,
340
In v i e w
DAlg
and
between the
D
A19.
DAgg,
Not,
D algebra
where
(5.33)
=
D morphism
= (f,~,~')
(5.34)
~
with
strict
,
~* = (f,
):
a
last
exactly
which
the
D A ~
of
ones
they
some are
of
(those
implicitly.
isomorphism
~'
§5.4,
are o n l y
A corresponds
.k'A,
between
paragraph
we do k n o w D
:
=
isomorphism
D morphisms
we g i v e
a
an
the
we k n o w
§4.6,
a'
F
in
and h e n c e
.kA,
a
,
The
we h a v e
we s a w
A with action
(A,a,a',a')
a
as
but
~ = i and f' = i), In a n a l o g y
The
then,
and D *  A g g , ; the
Dmorphisms;
strict
with
of §5.4,
=
a
A ~ B c o r r e s p o n d s
explicitly.
to
the D  a l g e b r a
.kA.
to the
Dmorphism
where
~* . k A ,
=
>'
* f .k'A;
=
and we have
(5.35)
>*
=
!
y:
f ~ g
if and only
if
>
=
i
and
>'
=
I.
Finally,
(5.36)
is a D  2  c e l l
We can m a k e every
Dalgebra
D action extend objects
this
statements
(A,a,a',~')
, and
so on.
to the m o r e
A of K but
our proof; done.
a
these
we l e a v e
is of the
The
same
general
2functors the
if and o n l y
easy
A:
in the a l t e r n a t i v e form
(5.33)
techniques
case
where
as
form
that
for a u n i q u e in §4.7
the a l g e b r a s
L ~ K, w h i c h
details
if it is a D  2  c e l l .
we n e e d
to the r e a d e r ,
and
allow are
us
to
not
to c o m p l e t e suppose
this
341
5,6
In this
prove
our
second
analogous
main
to that
proceeding avoid
section
endo2functors
Theorem
5.1
2complete doctrines DD'
2categ0ry
Define
(5.38)
K,
sj
h:
=
and ~h
l.atter b e i n ~
is not could
but
state
and
in d e t a i l
make
it so by
as we give
it we
direct.
"underlying
objects"
of all
our a l g e b r a s
be a p s e u d o suppose
distributive
that m a p s
s:
D
~
DD'
from
law on the
(D,D',p,w)
(D,D',p,w)
b__e the u n i q u e
~
D*.
into Then
strict
jj '
DD'
= 1.
one
to
that
DD' * D
as t h e
~ D D
sh = i, and
manner,
by K = (k,k',k):
Let
composite
, D
kk '
Then
and
are r e p r e s e n t e d
such
less
(D,D',p,w)
is a D  a l g e b r a .
(5.37)
the
itself
doubtless
direct
we p r o c e e d
of K.
Let
D morphism
4.1;
by b e i n g
section
to §4.8)
The proof
is a m o r e
diagrams
In this are
theorem.
of T h e o r e m
as t h e r e
large
(analogous
.
m
there
is a m o d i f i c a t i o n
This
appl.i.e.s to b o t h
the
that
where...the t h i r d
element
n:
I ~
hs
such
lax and..the p s e u d o qf a m a p
from
that
sO = 1
cases
 the
(D,D',p,w)
t__oo
^
a doctrine
is required........t..obe an isomorphi.sm;
.isomorphism ~ and
so is q.
in the
latter
case
k is an
342
Proof case
As in the p r o o f take
below
care
(h,h)
of
thus
§3.5
the d e t a i l s
below
D
as
.kD
D'
is a D '  a l g e b r a
,
a'
Dalgebra
on it, n a m e l y
By
the u n d e r l y i n g
b
(DD',
because
in the p s e u d o
whence
~'
below
we use
the a' are
the
stronger
n is i n v e r t i b l e
b, b',
DD'
were
action
from
m
, and
(5.33)
we h a v e
=
m
,
.k'D
with
action
with
Daction of DD'
hence
actions
on D
and on
A and B r e s p e c t i v e l y .
by
object
b');
the v a r i o u s
names
with
where
=
Dalgebra
whence hand
so f i n a l l y
of n a m i n g
a
(5.12)
case;
if t h e i r
(D*,a,a',a')
m
below,
is i n v e r t i b l e
is a D  a l g e b r a
Dalgebra
the d i f f e r e n c e s
2.6.
the p u r p o s e these
4.1,
k is an i s o m o r p h i s m ,
(5.39)
in the p s e u d o
we treat
(5.39)
h
by
of T h e o r e m
For DD',
of t h e m s e l v e s :
is i n v e r t i b l e
invertible;
of T h e o r e m
a'
m',
is t h e r e f o r e
=
m
.kD
so we can
mD'(also
is DD'.
it is also
.
form
called
Thus
a
DD'
the free
b = (b,b')). is a
a D*algebra
with
action
. Since
statement
is the
free
of the t h e o r e m
satisfying
a unique
shall
D'
need
below
r D
D'morphism.
Moreover
is a strict
way
D morphism
s:
of the
D
~ DD'
composite
from
with
s is a f o r t i o r i
the d o c t r i n e a'
on D
j':
whence
map
given
the d o c t r i n e
D'morphism;
composite
the v a l u e
D morphism,
the D '  a c t i o n
in the u s u a l
by its
is as in the
s
As a strict
that
strict
on i, t h e r e
~ DD'.
k'
note
D algebra
(5.37).
We
(5.40)
D
I ~ D'
by
map
as in
k'
is a s t r i c t (5.39)
k'. (2.4)
But
a strict
k'j'
Thus and = j
D'morphism;
is just
that
arising
the c o m p o s i t e (2.5)
(5.40)
it is d e t e r m i n e d
since
k'
is a
~3
doctrine may
map,
so that
be w r i t t e n
using
(5.37)
we h a v e
~ D'
too
composite
(5.41)
(5.40)
and
(5.42)
(2.5)
H.jD'
and g i v e n
(5.43)
D'morphism
is g i v e n
=
Since (2.4)
D
is a D  a l g e b r a a unique
=
k':
strict
D' *
we w r i t e
and Dk'
Substituting that
h has
We
Dmorphism.
is
• D
is a free H:
one,
DD' ~ D*
there
is by
satisfying
.
h
by
(5.14),
[u'
=
(5.38)
being
a = (a,~') strict
as
~.'.D'Dk'.
composite
in the
of a f r o m statement
(5.39),
strict
Dmorphism
~ ~ DD'.
~ D s
we
see
of the theorem.
s is a f o r t i o r i
~ D'
k' a l r e a d y
it
that
D morphism,
the
Because
As a D '  m o r p h i s m ,
for h the v a l u e
the v a l u e
a strict
f o r m H = (h,h,~').
(5.25).
= 1,
value
Consider
extended
(Dk',l)
conclude
in this indeed
DD'
Dmorphism
h = I by
a.Dk',
Being
the
D*
H in the
as in §5.3;
=
that
a
Dmorphism,
h
and
~ DD
is a strict
(5.44)
we c o n c l u d e
as the c o m p o s i t e
DD'
a D'morphism.
§5.2,
jD'.
explicitly
§5.3,
by
by
Dk ' As in
which
JD'
is a strict
sk'
= jj',
~ DD'.
j'
jD'
= sj
as
I
Since
sk'j'
a strict
344
As in By
(2.4)
(5.42)
another
and
and
name
(5.45)
(5.41),
si
:
Now
let H
(5.46)

now
D
~ DD'
prove
DD' ~ D
composition,
and
H.
since
Since
(5.17)
is
this
s corresponds
(5.37)
the
of §2.3,
r:
with D
i ~ DD'
,i r e p l a c i n g
corresponding
to
D,A
to
So
=
h .jj '
=
m
.kk' .jj'

m
.j j
=
m
.D j
=
j
by
since
(5.38) k,k'
m
.D j
in the s p e c i a l
case
(5.47)
h
.D j j ' . h
result
The
of the p r o o f
=
maps
.j
since
reflexion
are d o c t r i n e
now
=
of
follows
I
§2.3
(doctrine
in w h i c h
from Theorem
that
rest
by
be the D  m o r p h i s m
to the D  m o r p h i s m
situation
is jj'
we are
Our d e s i r e d
which
jD'.
i.
By
hr
Therefore
is jD';
with
we h a v e
in the
respectively. s:
(5.34)
preserves
(5.35),
So we are
its c o m p o s i t e
Hence
= (h,):
as in
sH
composite
by
I.
correspondence by
it is d e t e r m i n e d this
for JD'
corresponding
itself
(2.5),
i.
consists
in v e r i f y i n g
this.
axiom).
(2.23) 2.6,
commutes.
once
we
345
W e d o this,
(5.48)
by hindsight,
a' .D'JD
=
in s e v e r a l
small
steps.
First,
by
(5.23),
i.
Compose
this
w i t h D'k':
replace
D'jD
.D'k'
D'D' ~ D ' D
and use
by D ' D k ' . D ' j D ' .
Using
the n a t u r a l i t y
the t h i r d
of j to
equation
of
(5.44),
we n o w h a v e
(5.49)
h'.D'JD'
U s e the
second
(mentally so t h a t

equation
replacing (5.49)
(5.50)
h
i.
of
f:
=
h'
DD' ~ D ) ;
in t e r m s
we h a v e
mDD'
satisfied
,
,
~ DD'
,
Db
,
= D D DD'
* h h
D D
D *D *h
and u s e
D
*
(5.52)
(That
h
is,
on the
(5.50)
of
~
to
front
simplify
*
D D D
~ D
we h a v e u s e d
end w i t h
the
~
.m DD' .D k'jD'
and t h e n r e v e r t e d
.k'DD',
=
h
(5.50)
to l i n e a r
*
right
W
.D b
side.
D D'D' We
is
b ~ DD'
* h
D D
, ~ D m
~ D D DD',
end u p w i t h
*
.D k'jD'.
to k i l l t h e D h
rather
D* h
, , ~ Dm
D k'jD':
H
, ~ D DD'
Dh
m
sides
= h
by the D  m o r p h i s m
b
~ D DD'
D *h
both
h'
1.
D D DD'
Compose
of h
to
N o w o n e o f the a x i o m s
(5.51)
to e x p r e s s
A ~ B by h:
simplifies
.k'jD'
(5.34)
on t h e r i g h t
than diagrammatic
of
notation).
(5.51),
h
~8
Now be w r i t t e n
compose
(5.52)
with
kD'j':
~ D D'
, D D'j'
kD'
we t h e n
have
on the r i g h t
h
or h
.D y.kD'
this
is e q u a l l y
this
is I.
right
side.
which
say, h
.D b
where
.D
is
.k'jD'.D'j'
But
composing
Simplifying
D D'D'.
k'jD'.D D'j'.kD',
y = b
.kDD'.Dy.
So a f t e r
side
~
by
(5.34),
(5.52)
a little
what
with
By the n a t u r a l i t y h*.kDD' kD'j',
we get
= ~,
we get
on the
left
and by
h
.m DD' .kk'jj'
Write
kk'jj'

as the
DD'
side,
i.
composite
~ D D
, , D D jj'
m D D DD'
to r e p l a c e
m DD'.D
kk'
and the use D Jj'.m
(5.54)
which
.
Then
h
by
the n a t u r a l i t y (5.53)
.D J J ' . m
(5.38)
becomes
.kk'
is the
of m
=
(5.47)
i,
that
we
seek.
D jj'
of k, (5.44)
! on the
have
(5.53)
can also
as
DD'
What
DD' ~ D D'D"
by
we n o w
347
6.
6.1
P s e u d o  c o m m u t a t i v e doctrines and clubs
An endofunctor D of C a t
is m a d e into an e n d o  2  f u n c t o r by
giving a natural t r a n s f o r m a t i o n [A,B] subject to two axioms we derive a natural
(6.1)
t:
AxDB
~
~
[DA,DB]
(the "strength" of D)
(multiplicative and unitary).
From this strength
(and in fact 2natural) t r a n s f o r m a t i o n
D(AxB)
as the image under a d J u n c t i o n of the composite
(6.2)
A ~
[B,AxB]
~
[DB,D(AxB)]
.
It is well known that giving the strength of D is equivalent to giving t; in fact t and the strength are mates, [13]
§2.2, under the adjunctions
xA ~ [A,]
in the sense of
and xDA ~ [DA,] ; so
that t is often called the "monoidal strength" of D:
of course t must
satisfy two axioms c o r r e s p o n d i n g to those for the strength. If now (D,m,j) is a doctrine on Cat, it is easy to express the m o n o i d a l strength of D 2 in terms of t, while that of I is the identity.
The 2naturality, as distinct from the naturality, of j and
of m can be expressed in terms of the m o n o i d a l
strengths,
in the form
of a commutative d i a g r a m involving j and t and another involving m and t.
So all told, to make a mere monad
(D,m,j) on C a t
into a doctrine is
to give a natural t as in (6.1) satisfying four axioms. We use the symbol # in a general way to denote conjugation under the symmetry c:
(6.3)
t#:
DAxB
AxB ~
~ D(AxB),
namely the composite
BxA
of C a t .
So alongside t we also have
348
(6.4)
DAxB
D(BxA)
~ Bx]Zz~ ~
c
t
* D(AxB). Dc
We can then form the 2natural
(6.5)
~:
DA×DB ~ D(A×B)
as the composite
(6.6)
DAxDB
t~ D(AxDB)
Dr* D2(AxB) "*m D(A×B),
as well as its conjugate
(6.7)
d#:
DAxDB ~
D(AxB),
namely the composite
DAxDB ~t D(DAxB) D~# D2(AxB) ~m D(AxB).
(6.8)
Moreover,
using I for the unit category,
(6.9)
d ° = jI:
I ~
DI.
Kock [14]
shows that d and d ° enrich D to a monoidal
2functor
(D,d,d°):
monoidal~
the extra condition
(6.1o)
Cat ~ Cat.
This is not in general
needed
symmetric
for this is precisely
~ = ~#.
The unit j: natural
we can set
I ~ D is always monoidally
transformations
2natural"
there
and "symmetric
2natural;
is no difference
monoidally
2natural".
recall
between
that for
"monoidally
The multiplication
349 m:
D 2 * D is not in general m o n o i d a l l y
2natural, but is so if and
only if (6.10) is satisfied. Kock called a doctrine  or more g e n e r a l l y a Vmonad on a symmetric m o n o i d a l closed V  commutative Because of the last remark, call t h e m m o n o i d a l
Vmonads;
they are the m o n a d s on V in the
2category of symmetric m o n o i d a l
algebras V D is symmetric closed,
U:
V D ~ V, F:
and n e c e s s a r i l y lie in the
Vcategories.
completeness assumptions
if it admits coequalizers;
For commutative D,
on V, the Vcategory of
and indeed symmetric m o n o i d a l closed
and the forgetful and free Vfunctors
V ~ V D are symmetric monoidal.
if vD,u,F are monoidal,
(6.10).
it has now become rather more common to
2category of m o n o i d a l Vcategories,
nnder the m i l d e s t
if it satisfied
so is D = UF.
Conversely,
of course,
Thus, m o d u l o the a l w a y s  t r o u b l e 
some m a t t e r of the existence of coequalizers in V D, c o m m u t a t i v i t y of D is the n e c e s s a r y and sufficient c o n d i t i o n for vD,u,F to be symmetric m o n o i d a l closed:
it generalizes Linton's c r i t e r i o n
(cf. [61 p.549)
when V = Sets and D comes from a finitary theory. A final remark at this level. monad on Sets.
Suppose D is a commutative
Then, being symmetric monoidal,
D takes a commutative
m o n o i d A to a commutative m o n o i d DA, and in fact lifts to a m o n a d D on the category of commutative monoids.
So if D' is the monad on
Sets whose algebras are commutative monoids, we get an honest d i s t r i b u t i v e law p: 6.2
D'D ~ DD'
In the doctrine case,
c o m m u t a t i v i t y as expressed by (6.10)
seems to be rare in natural examples.
We call a d o c t r i n e pseudo
commutative if there is instead an i s o m o r p h i s m (invertible m o d i f i c a t ion)
satisfying suitable axioms y#y:
(one of which is that 7 be involutary;
d ~ d# ~ d is the identity).
Then we get a p s e u d o  d i s t r i b u t i v e
350
law
(p,w):
D'D ~ DD'
the last paragraph) the d o c t r i n e
as in §§5.1,
the m o n a d
for symmetric
for in all m y everything
examples
becomes
I succeed
easier
a more
sketch
this
because
compact
structure
but
in the above
Cat/~,
(as in
rather
would
treatment
generality,
and then
all the diagrams
come down
be pleasant,
but unless
of it, my a p p l i c a t i o n s
the extra complication. that
and let D be p s e u d o  c o m m u t a t i v e
satisfying
monoids,
for a club D in
Let then A be a D'category, category,
now D' is not
categories.
The extra g e n e r a l i t y
in f i n d i n g
do not justify
than
D is Do
much
by one dimension.
for c o m m u t a t i v e
monoidal
I shall do no more
5.2, where
axioms to DA.
is, a symmetric
with y as in (6.11)
to be determined.
We give
Its tensor
and
product
monoidal
a symmetric
identity
object
monoidal are given
by
(6.12)
DAxDA ~ D(AxA) d
(6.13)
I
~ DI ~ DA; d° DI
its a s s o c i a t i v i t y those
~ DA, D@
and right
for A (called
identity
isomorphisms
a and r) by composing
DAxDAxDA
~~ dxl
are obtained
from
Da with
D(AxA)xDA ~ D(AxAxA) d
and Dr with
DAxI
~
ixd °
Its c o m m u t a t i v i t y as the composite
DAxDI ~ D(Axl). d
i s o m o r p h i s m is obtained
from that
(called
c) for A
351
(6.14)
DAxDA
DAxDA
...........
a
T'
d
Dc
D(AxA) ...........
" D(A×A) Dc DA
where y' denotes because
Dc.y.
The m o n o i d a l  c a t e g o r y
(D,d,d °) is monoidal;
(c 2 = i and the hexagonal involutory
YAxB,C
the symmetricmonoidalcategory
axiom)
require respectively
and that it satisfy a kind of hexagonal
to ~A,C and
axioms
that y be
axiom
( relating
YB,C ).
The above passage monoldal
axioms are immediate
from A to DA respects
functors and the corresponding
strict
symmetric
natural transformations,
and therefore gives us a lifting of D as in (5.1), whence also a lifting as in (5.2).
The lifting of j at the
automatic,
because
symmetry:
for this it suffices
of y with jxl:
(D,d,d °) is monoidal,
(5.2) level,
(but not strict)
that is, to enrich
symmetric monoidal
functor
here I use m in place of the more usual m, which
unfortunately triple
the
AxDB ~ DAxDB is the identity.
D2A ~ DA to a strong
(m,m,m°);
except as regards
to impose the axiom that the composite
It remains to lift m at the m:
(5.1) level is
has another
(m,m,m°).
sense in §5, being in fact the name of the
We can take m ° to be the identity.
To get an m,
it turns out that we need a m o d i f i c a t i o n ~:
d(mxm) ~ m.Dd.d:
D2AxD2B ~ D(AxB).
t,m,j allow us to write this as ~: we set p = m.Dt.m.D¥.t #. a symmetric monoidal
The four axioms satisfied by
m.Dt.m.Dd.t # ~ m.Dt.m. Dd#.t #, and
Of the three axioms saying that
functor,
(m,m,m °) is
the one involving m ° is satisfied
in virtue
352
of our a x i o m ~(J×l) stating
that
satisfied three
(D,m,j)
in v i r t u e
more
axioms
distributive I
that
or at m o s t
reluctant
to d e l a y
I have
on to the
not
this
club
the n o t a t i o n
F:
D ~
letters
succeeded longer
one.
So
pseudo
of T ( S I ( R I I
laws
T[S I
The
similarly
functor by
t:A
x
~ for rS, we
shall
image
similarly the
club
and
case
we
and we of D
so on;
quite
OoD
... S T ) and under
j:
2natural similarly
law for m e x p r e s s e s ... RT~T))
T(~
ordered
are w r i t t e n of
for
J ~ D of the
for m o r p h i s m s ;
The
use
or
~ D
the
is
in fact
generally
denoted
equalities
DoB ~ D o ( A x B )
So I
set of a x i o m s
if we use m:
... S ~ ( R T I
of A×B
to do so.
as for p e r m u t a t i o n s
S T ] to T(S I
for m o r p h i s m s .
and the m o r p h i s m s
a n d am
the a u g m e n t a t i o n
associative
... RI~ I)
that
them,
In the
to date m has
the
to
initio.
as w e l l
in [ 9 ] § 1 0 ,
... RT~T) , and
trying
case
for a club D in C a t / ~ ,
that
...
club
a definitive
no c o n f u s i o n
for m and J e x p r e s s
= T, and
while
T for FT,
although
the
for ~ r e d u c e
in so r e d u c i n g
law ab
numbers,
cause
as
with
In p a r t i c u l a r
of J is ~ E D.
T(S I ... S T ) ( R I I
to be g i v e n
j are
our d e s i r e d
doctrine".
of s y m b o l s
sends
Again
object
The
axioms
= i, l e a v i n g
we n o w have
giving
D = Do
write
of D,
then m
the e q u a l i t y
that
for n a t u r a l
for m o r p h i s m s .
objects
we get
three
involving
y(j×l)
by a n a l o g y
commutative
so short
It should
DoDo ~ Do;
the
yet
of [ 9]§10.
the m u l t i p l i c a t i o n
~(T)
two
six a x i o m s
case w i t h o u t
5; we t y p i c a l l y
we are r u n n i n g
unitary
that
volume
suppose
use
functions.
the
axiom
the p s e u d o d i s t r i b u t i v e
We n o w
unique
the
three,
for the y of a " p s e u d o
Greek
Of the final
law.
but
6.3
same
for ~ e n s u r e
follows:
establish
two.
is a d o c t r i n e , of the
suspect
one or two,
pass
= I, l e a v i n g
(6.1)
with the
two
... 2) = T a n d pairs
(A,B)
that
are
and ( f , g ) .
is e a s i l y
seen
353
t( A, S[B 1 ... B ~
t
= h[
for t #.
× DoB ~ ~o(A×B);
they
...
A,B>,
l;
H e n c e we c a l c u l a t e send the object
{ T [ A I ... AT] , S[B I ... B ]) r e s p e c t i v e l y
to
(6.15)
T(S
... S ) [ { A I , B I)
... ( A I , B a)
... ( A T , B I}
... (A
(6.16)
S(T
... T)[{ AI,B I)
... { AT,B I)
... ( AI,B ~)
... { A ,B ) ] ,
with similar
effects
and T occurs
~
(6.15)
on m o r p h i s m s .
times
in (6.16);
the l e x i c o g r a p h i c a l
lexicographical
order
(6.17)
zTS:
T(S
component (6.16).
of
... S) ~
(6.15)
(6.11)
S(T
is in the
is to give for each
... T)
is the p e r m u t a t i o n
and
(6.16);
of T~ d e m a n d e d
by a
then the ( T [ A I ... A T ] , S [ B I ... B ])
is the m o r p h i s m
By the n a t u r a l i t y in
,B B)
isomorphism
of y = YA,B
commutativity
in (6.15)
of the (B,~} •
w h o s e type FzTS = ~T,a comparison
of the ( A
(Y
o r d e r of the (~,B) , and in (6.16)
To give an i s o m o r p h i s m ¥ as in T,S E D a n a t u r a l
,B )], a
T
Here S o c c u r s T times
the order
T
zTS[I,I,
of zTS we m e a n
...,I]
of c o u r s e
from
(6.15)
that we have
to
354
zTS,
T(S...S)
(6.18)
S(T...T) l g(f...f)
f(g...g)
S'(T'...T').
T'(S'...S') ZT, S ,
(This diagram
is more complicated
and Ff = ~ is a function morphlsm
than it looks:
from T to ~'; the ~th g in f(g...g)
from the ~th S in T(S...S)
in accordance good sense,
in general T' 6 T,
with the conventions
to the @~th S' in T'(S'...S'),
of [9]
§i0; however
in §6.2;
it remains
only to put such axioms
those we require
for y.
case,
¥ and referring
forgetting
detail what we barely 6.4
However
outlined
we deal henceforth
with an isomorphism
~T,~ described
above,
one axiom:
namely
on zTS as ensure only with the club out in
in §6.2.
Cat/S together
natural
situation
only to zTS , and carrying
We define then a ~seudocommutative
in addition
it does make
both legs having the same type.)
Thus does the club case fit into the general sketched
is a
c!ub to be a club D in
zTS as in (6.17),
in the sense of (6.18), the c o ~ u t a t i v i t y
of the type and satisfying
of
zT,S(RI,. (6.19)
T(S(RI...R~)...S(R1...R))
"R~S(R1.)..R)(T...T)
I T(S...S)(R I...R~...R I . . . R ) zTS(I,I , . . . ,i)
S(T...T)(RI...R1...~...R )
S(RI(T...T)...R (T...T))
I
S(ZRI T,.,z R T )
S(T(R I...RI)...T(R
...R a)).
355
We derive some immediate consequences of (6.19). put S = ~ (so that c = I) and R I = ~.
First,
The top and the left edges are
then both the i s o m o r p h i s m ZT~ while the right edge is Z~T , whence
(6.20)
z~T = 1.
Now in (6.19) put T = i, S = i, R I = R.
(6.21)
zR&
Using
i.
Finally just put S = ~, R I = R, T arbitrary.
(6.22)
(6.20) we get
Using
(6.21) we get
ZRTZTR = I.
Hence z is involutary and satisfies
(6.20).
Before giving examples we make the following remark.
Any
club D becomes a strict m o n o i d a l category, with ~ as identity, set T@S ~ T(S...S).
M o r e o v e r F:
if we
~ ~ ~ is a strict m o r p h i s m of strict
m o n o i d a l categories when ~ is given the c a r t e s i a n m o n o i d a l structure T®~ = T×~ ~ ~ .
For a p s e u d o  c o m m u t a t i v e club, zTS:
symmetric monoidal;
T@S ~ S@T makes
the hexagon axiom is got by setting
R I = ... = R~ ~ R in (6.19), and the other axiom is (6.22). F:
Moreover
V ~ ~ is then a strict symmetric m o n o i d a l functor, as ~T,c is
the classical symmetry on ~.
This o b s e r v a t i o n is useful in limiting
our search for examples. Example 6.1
The club ~ itself.
~i + "'" + cT while ~ is i. the p e r m u t a t i o n ~T,~: satisfy the axiom make sense,
We recall that T(al,...,~ ~) is
We take zT
T~ ~ ~T of §6.3.
:
T(c...c) ~ C(T..T)
to be
This is easily seen to
(6.19); in fact if it did not the axiom would not
its two legs being of different types.
Note that an
~algebra is just a category with s t r i c t l y  a s s o c i a t i v e finite coproducts.
356
Example
6.2
coproducts.
The club S w h o s e algebras are c a t e g o r i e s  w i t h  f i n i t e The augmentation
define zTS (as we must)
r:
S ~ ~ is an equivalence,
and we
to be the unique m o r p h i s m such that
FzTS = ~T,~" Example 6.~
The subclub ~ of ~ with the natural numbers as objects
but with permutations as its only morphisms;
the a u g m e n t a t i o n ~ ~
is of course the inclusion.
"
Again z
T~
= ~T
,c
The Palgebras are 
the strict symmetric m o n o i d a l categories.
Example 6.4 categories.
The club P whose algebras are the symmetric m o n o i d a l By Mac Lane's original coherence result [23],
a u g m e n t a t i o n F is an equivalence of P with F(P) the unique m o r p h i s m with FzTS ~ ~ ,
Example 6.~
= ~ C ~.
the Again zTS is
.
The full subclub of ~ determined by the objects 0 and
i; it is the arrow category ~.
Its algebras are c a t e g o r i e s  w i t h  a n 
initialobject.
Example 6~6
The full subclub of ~ d e t e r m i n e d by the objects
it is the discrete category 2.
0 and i;
Its algebras are c a t e g o r i e s  w l t h  a 
distinguishedobject.
Example
6~7
Our r e m a i n i n g examples are all of the kind where the
a u g m e n t a t i o n F:
D ~ ~ is the constant functor at i E ~.
A club of this
kind is nothing but a strict m o n o i d a l category D, with T(S) = T~S and with ~ as the identity for @.
It is p s e u d o  c o m m u t a t i v e
it is symmetric monoidal,
in which case we set zTS:
equal to the symmetry c:
T~S ~ S~T.
Dalgebra
Since DoA
T(S) ~ S(T)
is just D×A,
is a category A with a strictly associative
action DxA ~ A
if and only if
and unitary
(as for example E acts on K in §3.2 above).
the club whose algebras are c a t e g o r i e s  b e a r i n g  a  m o n a d , is not an example,
for the m o n o i d a l
category ~ is not symmetric.
a
Note that
namely 6'
structure on the simplicial
We pass to some p a r t i c u l a r cases of this
~7
example.
Example 6.7,1
Let C be a symmetric m o n o i d a l category and consider
the club D whose algebras are categories A together with a coherently (but not strictly) instance,
associative and unitary functor @:
any tensored C  c a t e g o r y A is such an algebra
C×A ~ A.
For
(where by
C  c a t e g o r y I m e a n "category enriched over C "  a remark n e c e s s i t a t e d by the fact that I sometimes, Dcategory).
for a club D, call a D  a l g e b r a a
Since all the operations on A are unary, the a u g m e n t a t 
ion of D is constant at i.
We have to show that D is symmetric, and
is thus a p s e u d o  c o m m u t a t i v e club. the canonical strict m o n o i d a l
But we know just what D is:
it is
category equivalent to C; its objects
are nads ( C I , . . . , C n) of objects of C, its tensor product is given on objects by concatenation,
and its m o r p h i s m s ( C I . . . , C n } ~ ~ BI,...,B m}
are the m o r p h i s m s CI®(C2@...(Cn_I@Cn) ) ~ BI@(B2@...(Bm_I@Bm) ) in C. It is now clear that D is symmetric when C is.
ExamDle 6.T.2
Let D be the discrete category ~ of natural numbers,
which is symmetric m o n o i d a l with + as its tensor product. is a category A with an e n d o f u n c t o r E:
Example 6.7. 3 monoidal
A Dalgebra
A ~ A.
Let D be the discrete category 2 with the symmetric
structure having the usual m u l t i p l i c a t i o n of its objects 0
and i as tensor product.
This is not the same as Example 6.6, for
the a u g m e n t a t i o n is now constant at i, whereas there it was r0 = 0, FI = i. E 2=
An algebra is a category A with an endofunctor E such that
E.
Examole 6.7.4 monoidal
Let P be the arrow category 2 with the symmetric
structure given on objects as in the last example.
This is
again different f r o m Example 6.5, as the a u g m e n t a t i o n is again constant at i.
An a l g e b r a is a category A together w i t h an
indempotent comonad.
R e p l a c i n g ~ by ~op gives another example,
an algebra is a category bearing an indempotent monad.
where
358
6.5
Let D be a p s e u d o  c o m m u t a t i v e club as in §6.4, giving the
doctrine D : go_., let D' be the d o c t r i n e Po whose algebras are symmetric m o n o i d a l
(s.m.) categories.
We exhibit a pseudo d i s t r i b u t i v e
law of D' over D, as described in §§5.1,
5.2.
(We could equally
produce a pseudo distributive law if we took D' to be the doctrine ~ofor strict s.m. categories;
this requires only the easy o b s e r v a t i o n
that if the s.m. structure on A is strict,
so is that we construct
below on DoA; we shall not refer further to this case.) The first step is to produce the lifting extension
(5.2).
Let
(A,@,l,a,r,c)
DA = DoA a s.m. structure to get
(6.23)
D
(5.1) of D, and its
be a s.m. category; we give
= (DoA,~,~,~,~,~).
We set
T[ A I ... A T ] @S[ B 1 ... Bo]
= T(S...S)[ A I @ B I , . . . , A I @ B o , . . . , A T @ B I , . . . , A T @ B o] ,
and define @ similarly on morphisms.
We define the rest of the
structure by
(6.24)
(6.25)
[ = l[I] ,
(T[A I...]@S[B I...] )@R[C I...] ) ~ T[A I...]@(S[B I...]@R[C I...] )
T(S...S)(R...R)[
(6.26)
TIAI...I~
IF
(AI~B?)~C 1 . . . . I  ~ ( S . . . S ) ( R . . . R ) 1[ a , a , . . . , a l
Y
T[ k I ... ]
II T [ A I . . . I ® i[I]
Jl T[ AI@I .... ]
T[ A I . . . 1 ,
i[ r,r ..... rl
II
[AlO(BI®C l) . . . . 1,
3,59
(6.27)
m S[B I...]@T[A I...]
T[ A I. . .] @S[ B I. ..]
fl
H
S(T...T)[B10A1,...]
T(S ....S)[ AI@B 1 .... l
zTS[ c , c , . . .
.
,c]
The axioms for a,r,c follow from those for a,r,c together with
(6.22)
(needed for ~2 = l) and the special case of (6.19) where R 1 = ... = R~ = R (needed for the hexagonal (Note that when A is the unit category on Dol, %:
= P itself,
described
A ~ B is a strict
TIA1...ATI mations.
^
functor
structure If
RoA ~ R o B
sending
for s.m. natural
transfor
(5.1) of D.
(5.2) sends the nonstrict
o
(~,¢,~),
to (¢,$,¢°)
extension
a and ~).
6.1 above.)
so too is DoS:
and similarly
Thus we have the lifting Its automatic
I this is the s.m.
just before Example
s.m. functor
to T[¢A1...$AT],
axiom connecting
s.m.
o
where ¢:
~A ® CB * ~(A@B)
where ¢ = Roe, where S h a s
and where ¢°= ~I~°].
and ~ : I B ~ ¢IA,
components
T(S...S)/$ ..... 2],
(Recall that we are using $ for the more
familiar ¢ to avoid confusion
with the m etc. of §5.)
sends the s.m. natural transformation with T[A1...AT]component The functor joA:
~:
Simiarly
(~,$,~°) ~ (@,~,~°)
to that
TaA1,...,~AT]. A ~ DoA is the funetor
and is clearly a strict s.m.
functor because
z~
sending A to SIAl, = 1 by (6.20).
This
gives us the j, and the j of §5.1. The next thing is to enrich moA: functor mA = (moA,~A,m°A), m,m,m
(moA
)~ =
Since ~ = ~[~] l[II
~ poA
to a strong s.m.
as in (5.18); we abbreviate mA, ~A, m°A to
where no confusion is likely.
on D o D o A .
RoDoA
= l[![Ill
= ~; so we take m
o
Use ~ etc. for the s.m. structure = ~[~][II , we have
= 1.
Now let
360
X = T[ PI [ All...AI~I]
, . . . , PI:[ ATI" • .ATw ] ] T
Y = S[RI[BII...BIo I] , ...
be objects (moA)X
of poNoA.
We need m:
= T(PI...PT)[AII...]
(6.28)
(moA)X ~
(moA)Y ~
(moA)X @ (moA)Y
T(MI...MT)[AII®BII
)...S(RI...Ro))[AII@BII .... ] ,
M I = PI(S...S)(RI...R
On the other
...RI...Ra).
hand,
xSY = T(S...S)[PI[AII...]~RI[BII...] =
T(S...S)[PI(RZ...RI)IAII~Bll
.... ] .... ! .... ],
so that
(moA)(X@Y)
=
T(S...S)(PI(RI...RI)...)[AII~Sll
= T(NI...NT)[AII®BII
where
(6.31)
.... 
for example
(6.29)
(6.3O)
(moA)(XSY).
and
T(PI...PT)(S(RI...R
where
, Rq[ B(~I " " "B°Po] ]
.... ]
for example
N I = S(PI...PI)(RI...RI...Ro...Ra).
.... ]
Now
361
We now take the X,Ycomponent
(6.32)
T ( Z P i s ( l , l .... ,i),
of m to be
..., Z p T s ( l , l , . . . , l ) ) [ l , l , . . . , i ] ,
where for example ZPis(l,l,...,l): Using the properties
M I ~ N I.
(6.18)(6.22)
of z, it is trivial if
somewhat tedious to verify that m = ( m o A , m , m ) for a s.m. functor.
satisfies the axioms
It is clearly 2natural in A by its construction.
Finally, using the same properties of z and a large sheet of paper, we easily verify that
(D,m,j)
satisfies the axioms for a doctrine.
Thus we have our d i s t r i b u t i v e law p,~ as in §5.2. We shall not need the explicit values of p and ~. however,
should be noted.
One point,
As a 2natural t r a n s f o r m a t i o n t
poDo ~ D~Po, p comes from a map PoD ~ DoP in CatJ~, that we may as well still call p; see for example T h e o r e m 8.1 of [9]. a l t h o u g h P and D are both clubs in Cat/~, not lie in Cat/~.
For instance,
the map p:
it sends ~[T,S]
However,
PoD ~ DoP does
to T(S...S)[®,®,...,@] ;
the type of the former is T + ~, while that of the latter is 2T~; so p does not commute with the a u g m e n t a t i o n £.
This is in contrast with
say §4.10 above, where we did not have to move out of Cat/~;
and it is
perhaps the first example to show that, even if we are only concerned with clubs in C a t / ~
we need the fuller theory of [9].
(This was
adumbrated in 510.7 of [9].) We sum up the result of this section in:
Theorem 6.8
I f D is a p s e u d o  c o m m u t a t i v e
club, the c o n s t r u c t i o n
above defines a pseudo d i s t r i b u t i v e l a w of D' over D = Do, where D' = Po is the doctrine for syrmmetric m o n o i d a l c a t e g o r i e s .
Similarly
if D' is the doctrine ~o for strict symmetric m o n o i d a l categories.
382
The examples in §6.4 of p s e u d o  c o m m u t a t i v e clubs now give examples of pseudo d i s t r i b u t i v e
6.6
laws, as promised in §5.1.
For the special pseudo d i s t r i b u t i v e laws obtained in this way, ,
the Dalgebras, and hence the D algebras, than that given in §5.3.
admit a simpler d e s c r i p t i o n
We work out from first principles what they
are. To give such an algebra B is to give a s.m. category B together with an action b: = (b,~,b°):
DoB
~ B that
DB * B.
So b is to be a s.m. functor
satisfies the axioms for a Daction.
(we are using B , b rather than A, a to avoid c o n f u s i o n with the a s s o c i a t i v i t y a.)
The s.m.
structure on D o B
described in §6.4, which we employ without At the level of ordinary functors, that b:
DoB
~ B is to be a D  a c t i o n
further comment. the action axioms tell us
on B in the ordinary sense.
write the image under b of TEBI...BT] E D o B T(BI...BT)
T(SI...ST)(AII...AT1...)
We
in the usual way as
E B, and similarly for morphisms.
express the equality of T(SI(AII...)
is of course that
The action axioms now
... ST(AT1...))
with
and of I(A) with A, together w i t h the
similar equalities for morphisms. Since joB: B ~ D o B
is the strict s.m. functor sending B to
i B, the u n i t a r y axiom for an action immediately forces b ° to be the identity:
(6.33)
b ° = i:
I ~ ~(I) = I.
Since m ° = I too, the a s s o c i a t i v i t y axiom for an action is satisfied at the level of °components. The only extra piece of structure,
then, that a D algebra B
has, over and above its symmetric m o n o i d a l structure and its Dalgebra
structure,
is the natural t r a n s f o r m a t i o n
363
(6.34)
T(AI'''AT)®S(BI'''B
c
)
IDTIAI...AT! ,S[BI...B] T T(S...S)(AI®BI,...,AI@Bo,...,AT@BI,..,AT@Bo), ^
which
we a b b r e v i a t e
^
to bTS or just
to b.
^
The axioms following
(6.35)
for
(b,b,l)
to be a s.m.
functorreduce
to the
three:
T(A1. • .AT)@I_(I)
T(AI...A
)®I T
T(1.
~)(AI~I ..A OZ)

T
H
r
T(AI~r...A ~I)
~
T
T(A
. .A ).
l(r,r .... ,r)
(6.36) a
(T(A I. .. )@S(B I. .. ) )®R(C I. .. )
T(A I.. • )@( S ( B l ...)@R(C I. . . ) )
T(s...s)b (i I~B1.... )°mCa"" )
T(A I . . . ) @ S ( R . . . R ) ( B I @ c
T(S...S)(R...R)((AI®BI)®C
T(S...S)(R...R)((AI@(BI®C
1 .... ) ~
I .... )
l) ....
l(a,a ..... a)
(6.37)
T(AI...)®S(BI...)
T(S...S) (AI®B 1 .... )
c
~ S ( B I . . . ) @ T ( A I . . .
_~
zTs(c,c ..... c)
)
S ( T . . . T ) ( B I @ A 1 .... ).
364 ^
The
component
of the u n i t a r y
axiom
for an a c t i o n
reduces
to
^
(6.38)
b~IA],~[B] :
Finally
the
reduces,
~(A)@~(B)
component
~ !(!)(A@B)
of the a s s o c i a t i v i t y
in v i e w of (6.32),
is I:
axiom
A®B ~ A@B.
for an a c t i o n
to
(6.39) T(P r •.P~ ) (An...)~(~. • .% ) (Bli. • •)
^
bTS
T(PI" '"PT ) (S(RI" " .R )...S(RI...Ro )) (AII@BII .... )
T(S..S)(PI(AII...)@RI(BII...) .... )
T(Pi(S...S)(~I...~..ml... ~) .... )(AliOBll .... )
T(S...S)(bPi~.)
T(S...S)(PI(RI...RI)(AII@BII .... ) .... )
T(ZPis(l ..... i) .... )(i .... ,i)
T(S(PI...PI)(RI..mr..~... ~) .... )(Am~li,...) where
in the topmost
a r r o w T(P)
stands
for T ( P I . . . P T ) ,
etc.
Now w r i t e
^
(6.40)
eB
T[AI...A I
abbreviating
(6.41)
for
e:
it where
bT[Ai...A I,~[B],
desirable
T(AI...AT)@B
to e TB
or e T or e, so that
~ T ( A I ® B ~ . . . , A T @ B ).
365
We show that b can be given in terms of e alone, the axioms on reducing to simple axioms on e. Write
(6.42)
e#:
B@T(AI...A T) ~ T(B@AI, .... B®A T
for the conjugate (6.43)
e # = T(c,c,..c).e.c
of e under c.
Then (6.37) gives
^
(6.44)
b__iT _ e # .
Now in (6.39) set each P
equal to I and set S = ~.
Since z ~
=
1
^
by ( 6 . 2 0 ) ,
(6.45)
we f i n d t h a t
bTR i s t h e c o m p o s i t e
T(AI...A T) @ R(BI...B p) le
V T(RI@R(B l..[Bp),i T(e#'
• .,AT@R(BI...B p) . , e#)
T(R(AI®BI,..,AI®Bp),...,R(AT®BI,...,A~B
0 ))
T(R...R)(AI@BI,...,AI@B p, .... AT®BI,...,AT@Bp).
Consider the following five axioms on e:
(6.46)
T(SI...ST)(AII...ATa )@B
=
T(SI(AII...)...S
(A I...))@B
SeT
eT(SI...ST)I
T
T(St(All...)~B . . . . S (Arl...)~e)
ST(esI, ...,esT) • (Sl...S~)(AII~B
.... )
=
T(SI(AII@B, •.. )...ST (A~I@B .... ) ).
366 (6.47)
el:
~(A)®B ~ ~(A@B)
is l: A@B ~ A@B.
(6.48)
T(AI'eTI ' ' A T ) @ I ~
T(AI@I ,...,A @I)
m T(A I...A T ). T(r,r ..... r) a
(6.49)
(T(A I...AT)®B)@C eT@l
T(AI...AT)@(B®C)
1
T(AI@B , . • •)@C
eT
T( (AIeB)OC .... )
T(a,a ..... a) (6.50)
T(AI@(B@C) .... ).
T(A I. ..AT)@S(B I. . .Bq)
T(AI@S(BI...Bo),...~"~ ' )
S(T(A I...AT)@BI,...)
T(es#.. .e~)I
I S(eT.e T)
T(S(AI@BI, •.. )...S(AT@B I .... ))
S(T(AI®BI,...)...T(AI®B~,...))
T(S...S) (AI®BI,...)
S(T..]T)(AI@B I .... ). zTs(l,l,...,l)
367
Proposition
6.9
Let B be a symmetric monqidal
galgebra for the pseudocOmmutative between natural transformations and natural transformations
e as in (6.41)
of §5.5
(lax case).
by supposing b• or equivalently Proof
We have already
by (6.40) then
satisfyin~
(6.35)(6.39),
(6.46)(6.50)
We ~et e from b by (6.40) and b from
A B with all this structure
for the doctrine D
There is a bijec%ion
b as in (6.34) satisfying
where e # is defined by (6.43). e by (6.45).
club 9.
cate~or~ and also a
is precisely
an algebra
We pass to the pseudo case
e, to be an isomorphism.
shown that if we start with b and define e
(6.45) gives us back b.
If on the other hand we start
with e and define b by (6.45)• then
(6.40) gives us back e; just put
S = ~ in (6.50), use
of)
(the conjugate
(6.47), and use
The only other point to be verified baxioms with the eaxioms. implies its conjugate (6.48);
(6.x) #.
(6.37) identical
Next we get
(6.46)
and using
(6.21);
and using
(6.38).
from
with
is the equivalence
Note that each of the eaxloms First,
then•
(6.50);
(6.35)
and (6.38)
(6.50).
(6.49)
from
given the eaxioms,
identical with
Finally
to get
(6.36).
(6.49) and
(6.49) # .
(6.49) we next get the special case
from this•
from (6.46) # • and from
The reader will have no trouble
(6.36) and
D algebras.
(6.39).
led
The
(6.46) and (6.46) # , (6.36)'
From this together with (6.36)" where R = ~.
(6.49) # , we get the desired in supplying the details.
The first four of the eaxioms admit conceptual ions, and we are
(6.47).
(6.36) by setting S = R =
As for (6.36) we easily get the special case
where T = R = ~ by using (6.46) and
(6.x)
is identical with
latter is easy using the definition (6.45) of b, using and using
of the
(6.39) by setting S = R I = ~ in the latter
and we get
It remains,
(6.21).
D
interpretat
to the following definitive description
of
368
Theorem
6.10
doctrine Theorem
Let D be a pseudocommutative
club and let D
arising as in §5.5 from t h e pseudo distributive 6.8.
category~
together with for each B E B a__nn
enrichment
of the functor
(@B,eB):
B ~ B, s atisfYin~ axioms to be given below.
abbreviate
the ~ Al...ATJcomponent
B eT[ AI...AT] :
@B:
B ~ B to an opDfunctor
T(AI...AT)@B
symmetry c of B produces
6.52)
(B@,e#B).
For f:
We agree to
~ T(AI@B,...,AT@B )
o_~f e B t go e T where convenient; and we observe
functor
law of
Then a D al~ebra is a D  a l g e b r a B that is also a
symmetric monoidal
(6.51)
be the
that conjugation by the
from e B an enrichment
of B@ to an
opD
The axioms now are:
B ~ C, the natural
transformation
@f:
@B ~ @C
is opDnatural.
6.53)
a:
(@B)@C ~ ®(B®C)
6.54)
r:
@I ~ i is opDnatural.
6.55)
e and e # are related b[ the c o ~ u t a t i v i t y it suffices @enerators
to impose ' for T,S belonging of the discrete
The above is for the lax c ase~ opDfunctor
is oRDnatural.
of (6.50), which to a set Of
c!u b IDI.
for the pseudo case we require the
@B to be stron$, tha ~ i s ~
e to be an isomorphism.
Proof
By [9]
§10.8 to say that
(®B,e B) is an opDfunctor
say that the e of (6.51) is natural satisfies [9]).
(6.46) and
The axiom
T and the A .
(6.47)
(corresponding
(6.53) and
to (6.49) and (6.48) respectively. Proposition
in T and the A , and that it to (10.25) and
(6.52) above just makes e natural
The axioms
consequence
in B as well as in of [9],
The result now follows from
of (6.50)
(6.46) and is a
for R,S and for P ,S.
(Qf. [9]§10)
description
using
(6.50) for T,S, where T = R(PI...Pp),
When ~ is presented
by a small number of generators
and
the above theorem gives a very compact
of D algebras.
6.11
categories,
of
6.9, except for the last clause in (6.55).
(6.19) we see that
Example
(10.26)
(6.54) reduce r by (10.27)
The proof of that last clause is easy:
relations
is to
Let D be itself the club P for symmetric monoidal which is pseudocommutative
by Example
6.4.
Then a
,
D algebra B has two symmetric monoidal denoted as above by (B,®,l,a,r,c), (B,$,N,a',r',c'). opmonoidal
be natural
(6.54) for r. three cases
(AI~A2)@B ~ (AI@B)@(A2@B),
eN:
N®B ~ N,
The axiom
for the objects
of 9, subject to the
(6.52) merely asks these components The condition
reduces as usual to two diagrams, Finally
(6.55) requires
(T,S) = (~,~),
the conjugate
e B of ®B to a symmetric
e~:
in B as well as the A s.
ops.m.natural
denoted by
we need as usual only to give its components
to the generators
usual three axioms.
the ~'structure
and the gstructure
To give the enrichment
functor~
corresponding
structures:
(~,N), and
of that for (@,N)).
(6.53) that a be as does similarly
us to impose (N,N)
to
(6.50) in the
(that for (N,@) being
All told we have ten simple axioms
370
involving the
e~ and eN; and
independent
axioms
it is easy
among
the
24 given
Laplaza
[18].
6.7
If we take the d e s c r i p t i o n
6.10,
and m o d i f y
at the
same time
a Dfunctor
it by c h a n g i n g changing
rather
the a l g e b r a s
for
In the
D'Alg
but
the sense
its name
to
some new doctrine,
situation
on the
are p r e c i s e l y
situation
of the a r r o w
e in
@B to
a description
w h i c h we shall
call
got
with D '  a l g e b r a s
of
D.
D rather
law we had c o n s t r u c t e d
2category
(6.51),
e enriches
we obtain
have
by
given by T h e o r e m
e, so that
of §5 we should
distributive
for this
of D algebras
than an opDfunctor,
if from the pseudo not on
to see that these
than D
a doctrine
as objects
and
,
opD'morphisms
as 1cells. Since ~ is invertible this D case , like the D case; there is n o t h i n g to change except the
exactly
is sense
,
of the
2cells.
co__reflexion
s:
In p a r t i c u l a r D ~ DD',
Theorem
with h:
5.1 in the
DD' ~
Dcase
D, sh : i, ~:
gives hs ~
a
i,
sn = i, n h : i. The point
of our i n t r o d u c i n g
it is of the form general
Do
for a club
p r i n c i p l e s of[ 9]§i0,
this
D is that,
Cat/~.
D in
is clear
in the lax case, In the light
from T h e o r e m
6.10;
of the for
,
the type
of the natural The
pseudo
D for the pseudo
cases,
type of e in since this F:
D ~
transformation
sometimes (6.51)
arise
case,
in ~.
and the D
from clubs
is in ~, w h i c h
in
for the lax and the
Cat/~,
only happens
is to be so for all T E D,
~ is constant
e lies
namely
when the
when T = i in (6.51);
it only happens
when
at i.
We can bring Example 6.11 above,
the one
studied
by Laplaza,
,
under effect
the
Dsetting,
by passing
just by r e p l a c i n g
to the opposite
because B °p is symmetric
monoidal
the m o d e l B by B °p (i.e.
doctrine). when B is.
This
works
in this
in case
371
Example
6.12
Another
Dsetting. categories also
being maps
Let D be the club with finite
symmetric
object
important
f~(1)
is a coproduct f :
naturally
6.2,
C o n s id e r
For T E S with
for suitable
occurs
S of Example
coproducts.
monoidal.
T(AI...AT)
example
whose
in the
algebras
are
any S  a l g e b r a
B that~is
rT : T, and for A
E B, the
in B of the A
, the c o p r o j e c t i o n s
~ ~ T in S whose
types
are the various
i ~ T in S. It follows
Salgebra.
that
there
is a unique
For the n a t u r a l i t y
in T of
e turning e demands
B into a
commutativity
in
eT T(AI@B'''''AT@B)
~ T(AI'''AT)@B
T
F eI
and
e~ is to be I by
~th component this does
(6.47).
is f (I)@I.
satisfy
So the only possible
On the other
hand
e is that whose
it is easily
seen that
the axioms. ,
We are coproduct, Po; in
led to the c o n c l u s i o n t h a t
in the c a t e g o r y
or equally
Cat/~,
6.8
that,
of doctrines,
in the category
S is the coproduct
We can apply T h e o r e m D
~
Dopo,
a reflexion
s:
equivalence
in the pseudo
the d o c t r i n e
of S and
of the doctrines
of clubs
case:
Cat/~
in
is the So and
and clubmaps
P.
5.1 to the present or sA:
So
D A ~ DoPoA,
or equally
situation, which
to obtain
is an
a coreflexion
,
s : s:
DA ~
*D
DopoA
which,
in the
lax case,
t a k i n g A = I, can be written
~ DoP. I shall
however,
observe
independently
just that
indicate this
briefly
special
of §5, taking T h e o r e m
how s is calculated.
case of T h e o r e m
First,
5.1 can be proved
6.10 as the d e f i n i t i o n
of
372
D algebra,
and a p p l y i n g
of the present of L a p l a z a ' s
§6 alone.
result;
"distributivity distributive
of Beck.
in (6.51).
The axioms
calculation
these
(6.50)
is in fact
is not
i when e = i; but easily
given by
satisfied
the pseudo
case,
*Dcategory.
Since
also
and only
s(A)
if s(A) does
expressed
in terms
* e; the functor particular
satisfied;
because
e # given
q with
we have The
and
by
(6.43)
of c in 9opoA
we make
it strict
o
is a D category;
(resp.
and equally D)morphism,
A, in the
even for
a we can write
free D *  c a t e g o r y
the usual more objects
T E 9; the m o r p h i s m s
D *A ,
case A commutes
of T h e o r e m
Then
flexible
if
results
of "9 are g e n e r a t e d by a,r,c,
all this
In the other
sq = i, qh = i.
independent
9oP)
and in the pseudo
to T(VI...VT).
a proof
(whose
are trivially
set e = i
so.
e to I.
sends T [ V I . . . V T]
D
"9 ~ 9oP respects
s: ,
sends
data;
of clubs.
and the objects
(h,h,h)
same
and of
We can therefore
for the lax case;
of any diagram
In the lax Dcase
^
At the
is
For T E 9 and for
in view of the value
s is a strict
out of the elementary
suitable
in §6.5.
of T ( A I . . . A T ) @ B
(not trivially
(and in p a r t i c u l a r
hence
at once the image
@,I,
whence DopoA
(6.27)). Thus 9opoA
made
given
to be equal.
(6.46)(6.49)
"(pseudo)
a free one,
structure
a short proof
between
and
in fact a free one.
an easy
shows
indeed
(6.51)
the techniques
quite
the c o n n e x i o n
sense
category,
using
for instance
sense of
is a 9  c a t e g o r y ,
T(AI®B,...,AT®B)
gives
in the
category, with the
AI,...,AT, B E 9opoA
2.6 directly,
it conceals
in the
PoA is a s.m.
time 9opoA
This
but
of ~B"
law"
also a s.m.
Theorem
those
structure,
direction
h:
sh = i and q:
by
of 9, and and
in
9op ~ "9 hs ~ i for a
To find q directly,
and to provide , 5.1, we enrich h to an op 9functor:
as a 9functor,
and as an o p  m o n o i d a l  f u n c t o r
o
it has h sense
= i and h formally
is now reversed).
the
same as the b of
The q is then formed
(6.34)
as in T h e o r e m 2.6
373
from the ~ of the resulting
op*Dfunctor
(h,~).
In the ease D = P studied by Laplaza, the composite
P ~ PoP with the equivalence
of s:
gives a funetor
it is easy to see that PoP ~ ~o~
rot:
P ~ ~o~ that is precisely his distortion
So as indicated
in §1.4 above,
theorem on page 231 of [19] •
our coreflexion
result
(Note that Laplaza
(191§2).
implies his
is considering
,
diagrams
writable
in *D  that is, involving
e but not its inverse

,
but studying their commutativity be an isomorphism.
in a model
He actually says "monomorphism",
have dualized would be "epimorphism", stands,
and is too rare in examples
Proposition
we observe
that,
i0) follows
all the m [A i...] A I,...,A m
since we
given in his earlier paper
from the same coreflexion theorem, f,g:
are different
and,
if
X ~ Y in
if Y = n[ml[All...],...,mn[Anl...]
co~ute
are different
then determined
which
to be worth pursuing.)
for a discrete A, two morphisms
po~oA necessarily
e happens to
but this is not enough as it
The other main result of Laplaza, (18]
in which
where
for each ~, all the
(for the permutations
expressing
f,g are
and equal).
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