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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
Springer-Verlag 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 theory--Congresses. 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] 74-19483
AMS Subject Classifications (1970): 18 A15, 18 A35, 18 A40, 18 C15, 18D05, 18D15, 18D99, 18E35
ISBN 3-540-06966-6 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06966-6 Springer-Verlag 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, re-use 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 Springer-Verlag 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
2-category
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
2-category.
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 2-monad,
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
2-categorical
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 2-categories.
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 2-category theory. inside any 2-category, 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 2-category - e s s e n t i a l l y the same thing as a finitely
2-complete 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
2-categories with a
structure so r i c h that we can imitate those arguments, Yoneda lemma, that depend upon the hom functor. 2-category 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 2-category 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. Kelly-Street: 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 2-categories
75
Ross Street
Fibrations
and Yoneda's lemma in a 2-category
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 epic-monic
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
V-based
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 epic-monic
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,
V-based 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 M-adjunctions.
completions
cribles. and m o n o i d a l
system
FACTORISATION
m in a V-category
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
section-headings
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 M-adjunctiqn.
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 "evaluatlon-at-singleton".
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,
set-based
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 M-monics
(i.e.
inverse
of M-monics
T: C ~ B is M - c o n t i n u o u s
M-images).
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 M-complete
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
M-continuous
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 V-adjunction
by [16|§4.1.
are b a s e d on the following:
Let C be an M-complete
full s u b c a t e g o r y
C C B is M-continuous. 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 V-category
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 V-based 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 Z-orth0$0nal
of f13])
if B(s,B)
if B is Z - l e f t - c l o s e d Now suppose proper
V-category)
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 M-adjoint.
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 M-adjunction.
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 M-complete
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 V-category
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 S-small
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 one-point 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 one-point 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 S-complete and admits all intersections of M-subobjects,
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 M-adjunctions.
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 "M-cribles" 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 "M-preatlases"
(|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 M-cribles 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)
fA|FA, 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 M-crible 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 V-category A and M: A ~ C is a left Proof. A(F,G)
(with respect to
~ ~ A
o
admits
V-adjoint functor.
For each pair of M-cribles 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 M-monlcs
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 M-monics,
and i n t e r s e c t i o n s of M-monics
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
M-crible 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 M-adjunction.
In other words, F: A °p ~ V is an M-crible 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 M-faithful
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 M-erible.
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 well-defined
V-category,
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 M-faithful
C.
Proof.
To prove that M is extendable,
M-cr~ble with t r a n s f o r m a t i o n ~F: KF ~ C(MF,C) M-faithful,
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
M-monic
K(A(-,A)) -- C(M(A(-,A)),C) ~ C(MA,C). This makes K(A(-,A)), M-crible
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. M-faithful,
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 M-faithful 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,
V-adjoint 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 M-subobjects
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 M-faithful 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 S--iT: 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(A@B)
,,- T S ( A ® B )
P(I@q')
A@PB
= n'
where
B
q' E E and m E M.
TS(I@q') : TS(A@PB)
P(A~PB) m
Thus an obvious
sufficient c o n d i t i o n for
P(l@q') 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 M-cribles is a normal reflective subcategory of |A°P,v] if C(M(A@-),C) ition I.i).
is an M-crible
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 M-subobjects,
sections of M-subobjects.
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; B@s 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
s@B E Z ° for each s E Z ° we have that B(s@B,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
M-subobjects
is the
A(-,
R~F)
adjoint
structure to M).
This
R and
A criterion
if A is closed
of M-subobjects,
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 M-faithful
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 M-complete 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 left-adjoint
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".
left-adjoint
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
(~,~): S--4E,
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), 1389-1392.
[2]
Applegate, H. and Tierney, M., Categories with models, Seminar on Triples and Categorical Homology Theory, Lecture Notes 80 (Springer 1969), 156-244.
[3]
Applegate, H. and Tierney, M., Iterated cotriples, Reports of the Midwest Category Seminar IV, Lecture Notes 137 (Springer 1970), 56-99.
[4]
B@nabou, J., Introduction to bicategories, Reports of the Midwest Category Seminar I, Lecture Notes 47 (Springer 1967), 1-77.
[5]
Day, B.J. and Kelly, G.M., Enriched functor categories, Reports of the Midwest Category Seminar III, Lecture Notes 106 (Springer 1969), 178-191.
(6]
Day, B.J., On closed categories of functors, Reports of the Midwest category Seminar IV, Lecture Notes 137 (Springer 197o), 1-38.
[71
Day, B.J., A reflection theorem for closed categories, J. Pure and Applied Algebra, Vol. 2, No. i (1972), i-ii.
[8]
Day, B.J., Note on monoidal localisation, Bull. Austral. Math. Soc., Vol. 8 (1973), 1-16.
[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), 421-562.
[11]
Fakir, S., Monade idempotente associ@e & une monade, C.R. Acad. Sc. Paris, t. 270 (1970), 99-101.
19 [12]
Fischer, H.R., Limesr~ume,
Math. Annalen, Bd. 137 (1959)
269-303. [13]
Freyd, P. and Kelly, G.M., Categories of continuous
functors
I, J. Pure and Applied Algebra, Vol. 2, No. 3 (1972), 169-191. [14]
Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory, Springer-Verlag,
[15]
Kelly, G.M., Monomorphisms, Austral.
[16]
Berlin, 1967.
epimorphisms,
and pullbacks, J.
Math. Soc., Vol. 9 (1969), 124-142.
Kelly, G.M., Adjunction for enriched categories,
Reports of the
Midwest Category Seminar III, Lecture Notes 106 (Springer 1969), 166-177. [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., F-localisations 1973, 405-438.
Springer-
Berlin, 1971.
and F-monads, 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
set-valued
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 over-all 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 V-universe"
in the case where all categorical algebra is based on a
fixed symmetric m o n o i d a l closed category V. " V-structure" large)
in
on any category [A,B]
This enables us to put a
of V-functors
V-category 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 base-category
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
(~,~): S--iT:
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(q@l):
S(B@B')
~ S(SB@B')
]
S(!@q):
S(B'@B)
~ S(B'®SB)
I
~ S(SB®SB').
}
S(n®D): S(B@B')
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(SB@A)
1
S(l®n):
S(A@B)
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 internal-hom
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
F@G =
lAB FA @ GB @ P(AB
G/F =
lAB [P(-AB),[FA,GB]]
category
form:
-)
(1.2)
(1.3) F\G =
f
AB
[ P(A-B) ,[ 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
A@B 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
2-dimensional,
of p r o m o n o i d a l categories
(P,Hom,I)
The usual
1-dimensional,
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(A-B),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. bicate@ories
The concept
of convolution
in the sense of B@nabou
[3].
can be extended to include If
A = {Axy ; x,y E 0bj A} is a V-bicategory, [A,V]~
is a biclosed
with hom-categories
{[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 V-pro-bicategory.
in this article to V-bicategories
exercise once this conceptual theorem for biclosed
of the "one-object"
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. result-with-symmetry
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 left-adjoint
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 V-fractions 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 V-universe".
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
W-Cat
internal
hom. Property
and dense,
V-Prof
(cf.
into
between
them,
closed
tensor
E:
is strongly
V ~ W Thus
consisting
is embedded
bicategory
V-Cat
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],
V-profunctors
and the embedding
2.
hence
monoidal
W-Prof
is fully
product
the biclosed
cogener-ating bicategory
of V-categories
(but not
fully
and
and
on morphisms)
in such a way as to preserve
composition.
Property
3.
completion
IA,VJ*
equivalent
to
V-promonoidal 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
W-promonoidal
§4).
V-category
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 W-category. 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 W-promonoidal
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.
V-categories 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(A-B), 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 -@-: 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)
T-algebras
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
tensor-product
® 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(A@B,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 =
u-To-1
holds and this is equivalent to @ being a V-bifunctor. 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
T-algebras.
If (C,~) is a T-algebra 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 T-algebras 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 T-algebra is e x p o n e n t i a b l e in Proof.
T h e algebra
Let
~.
(C,~) be a T-algebra.
Then the adjunctions:
32
B(-~B,C)
~ S[-,C/B)
B(B~-,C)
~ B(-,B\C)
provide adjunctions: B(T)(F(-@B),C) ~ B(T)(F-,C/B) B(T)(F(B@-),C) ~ B(T)(F-,B\C) where F: B ~ B(T) is the free-algebra
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 n@l
~, T(T(C/B)®A)
I
T 2 (C/B) ®B
N(~@I)
T(C/B)@B
,~,
¢((C/B)~B)
iTe
I
{C/B
C
TC
where ~C/B is the adjoint-transform the adjoint-transform T(C/B)
N(T~@I)
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 T-algebra
(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 T-algebras 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(C@TD)
Tk ~ T2(C®D) ~ TO
T(C®D)
T(TC@D)
.
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(A@B) 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 V-categories the category [A,B]
of all
V-category if V is suitably compl-
V-functors 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°P@B ~
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 Ab-dense 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 V-category 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 internal-hom 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
internal-hom
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 internal-hom
[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,FX@GY))-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,FX@GY)).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-).(FX@GY) 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-).(FX@GY) 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
internal-hom 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
internal-hom
|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 V-fractions
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(Z-I). 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(A@s,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
"Z-continuous" all
Hence
C(B@s,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(s@B,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(s@A,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
internal-hem
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 tensor-hom
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 Z-continuous
A form an exponentially
biclosed
then so do the Z-continuous 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 left-adj-
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
f-lb
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
pullback-along-f 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 llmlt-preserving extension along the comonadlc functor
C/C ~
for all
f* .
f, :
73
Suppose
B
is a
cs-category 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 A-stable 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
A-stable.
is stable if
Z
is
A-stable. 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.
M-relations in
B .
with the evident universal property.
Z
Then, on takirg Z = E, Because
SpanB
is closed under exponentiation in
In particular,
E-M
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 cat@gorie des espaces topologiques, C.R. Acad. Sc. Paris, t.262 (1966), 1389-1392.
[ 2]
B@nabou, J., Introduction to bicategories, Reports of the Midwest Category Seminar I, Lecture Notes 47 (Springer 1967), 1-77.
[ 3]
Binz, E. and Keller, H.H., Funktionenr~u~neinder Kategorie der Limesrgume, Annales Acad. Sc. Fen., A.I.383 (1966), 4-21.
[ 4]
Day, B.J., Relationship of Spanier's quasl-topological spaces to k-spaces, 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), 553-558.
[ 6]
Day, B.J., A reflection theorem for closed categories, J. Pure and Appl. Alg., Vol.2, No.l (1972), i-Ii.
[ 7]
Day, B.J., Note on monoidal localisation, Bull. Austral. Math. Soc., Vol. 8 (1973), 1-16.
[ 8]
Day, B.J., On adjoint-functor factorisation, Proc. Sydney Cat. Conf., to appear.
[ 9]
Fakir, S., Monade idempotente associ@e ~ une monade, C.R. Acad. Sc. Paris, t.270 (1970), 99-101.
[I0] Freyd, P. and Kelly, G.M., Categories of continuous functors I, J. Pure and Appl. Alg., Vol.2, No.3 (1972), 169-191. [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
f--J u and f'--Ju'
ality.
and then m e n t i o n the
2-category
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 2-category, available
in
a 3-category).
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
2-category,
(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.
2-categories,
self-contained
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 2-category
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 1-cells
f: A ~ B etc.,
or 0-cells and
A etc.,
arrows
or m o r p h i s m s
or
2-cells 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
2-cells,
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
2-cells
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 2-cell f
u
uf =
g this
2-cells,
from 2-cells f
we g e t
of
v
composite
2-cells
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
2-cell
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
2-cell 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
2-cell
composites
the
vy.6f:
2-cell
Bg.u~:
uf ~ uhg ~vg,
uf ~ v k f ~ vg.
Thus
we
as
s u c h as
no
2-cell
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 2-categories,
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. 2-cells
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 2-category
There
is CAT,
is the
and when
sub-2-category
Cat of small categories. For a m o n o i d a l V-categories, of E81;
V-functors,
a g a i n with
categories
the
themselves,
transformations The a 2-category
category and
V-natural
sub-2-category 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
2-category
transformations,
in the
V-Cat 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 hom-set
V-CAT 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 2-category natural
transformations will
1.4.
Besides
by 2 - f u n c t o r
oint;
see [141
do not given
use
and by
2-cells
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
CAT-category, determines
transformation:
Similarly
in
the category
a
This d e f i n i t i o n
or "2-adjoint"
assigns
terms,
of L, arrows
what
namely
V-adjunction.
its we shall
CAY-functor
for 2-adjoint,
in the more
domains
and composition. to each object
is not only natural nB.Df ~:
a 2-functor
of K to arrows
of L, p r e s e r v i n g
identity
(1.1)
of
: CAY-adjNote
general
that we
senses
to t h e m by Gray in [lO].
to objects
2-cell
definition
transformation.
"2-natural"
B:
an object
below.
2-natural
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
2-cells
the e l e m e n t a r y
closed,
CAT-valued
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 2-natural
A of K an a r r o w
but also
2-natural
EB
category;
2-cells
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
V-functor
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
V-natural
V-functor
transformations
E form not
V = CAT the 2-natural
in other words
from the
only a set but an object transformations
2-CAT is really a 3-category,
of
D ~ E form a i.e.
a 2-CAT-
82
category.
We
follow
transformations
Benabou
[i]
modifications.
in calling
morphisms
of 2-natural
A modification
p: q -* ~ :
K "* L ,
D ~ E:
also written
D
E assigns
to
each object
A of K a 2-cell
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
2-cells
this
2-category
The what
reverse dual,
Observe
Kelly feels
1-cells
2-category
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 2-category
K°P(A,B)
2-cells.
duality
thus we reverse
are
2-functors
D:
CAT-adjoint;
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
2-adjunction
endo-2-functors
it.
ordinary
the r e s u l t
it c o m e s
are
a 2-cate-
so t h a t
we
K c° f o r the o t h e r
so that
2-cells
but
not
the
1-cells
= K °pc°.
say t h a t they
K, we g e t
2-natural
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
non-elementary
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
V-adjunction to h a v e the u s u a l
i__n_nthe
L ~ K are
to the
2-natural
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;
2-category
2-adjoint
2-CAT,
so t h a t in t h e
sense
83
of §2.1 below. i. 5 .
A great many
notions
2-CAT 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
2-cells
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 "op-lax
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
"morphism-like"
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 2-categories,
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
willy-nilly
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 "pseudo-N"
(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
2-CAT is a 3-category.
and this b e c a u s e commute
in V-CAT
definable
is itself when
continues
and Roberts
a pseudo-2-cat-
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 2-natural
of Kelly and
below,
[15],
pseudo-monads
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 ~ @(A@B)),
here;
such lax or pseudo papers
ADJUNCTION
and
"strict
monoidal
in [19],
as we need are
IN A 2-CATEGORY
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
"2-natural"
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)
2-monad)
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
2-monads,
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
"quasl-natural"
This
themselves
in the present
A second example in place
are
degrees
of
The utility
equations
of
(2.1)
and
Walters.
f---4u: A ~ B in a 2-category
u: A ~ B and f: B ~ A t o g e t h e r
with
2-cells
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
V-natural usual
s the counit,
K is V-CAT
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
f--4u 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
2-cells
operations,
86
Proposition n',s':
f'--~u':
a bisection
(2.4)
2.1.
Let
A' ~ B'.
between
is t h e
D,s:
Let
2-cells
a: A ~ A'
and
b:
B ~ B'.
2-cells
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
2-cells
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
2-functors
transformation. ,- EA
"
DB ................
Then
in
~ is the
under
Proof.
~
EB
L. D
identity
,
and
let
identity ~A
DB
~
2-cells
~ 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 2-fu,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 2-cell
and
an a d j u n c t i o n
2-naturality
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
2-category
in K, on the
with
and
2-cells
the m u l t i p l i c equations
= ~-~t. has b e e n
simpler
given
aspects
and
by
then
K = 2-CAT.
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 t-algebra
v: ts ~ s s a t i s f y i n g = i,
s together
A).
s ~ s'
v-tv
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
= o-v.
t-algebras
forgetful
functor
~r:
t 2 r ~ tr~
tp:
t r ~ tr'
FA:
K(A,B)
t-algebra
with
t-algebras
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,
t-algebras:
~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
2-cell
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.
2-cells
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
v-t~,
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
t-algebras.
the unit
s ~ s' are m o r p h i s m s
~''T 2 = T-H, 2
domain
"t-algebra"
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 !
2-cell
case
to t h o s e
of t - a l g e b r a s
3.2
a
classical
restricted
t-algebra
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 2-cells
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
2-cells
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
~':
t-action
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 2-functor A ~ representable,
Alg(A,t)
from K °p to CAT is
so that
Alg(A,t)
(3.13)
~ K(A,B t)
(2-naturally in A) for some B t in K, called the object of t-algebras. 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 2-naturality 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 = V-CAT 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
V-category;
so in this sense a t-algebra is an object s of the
V-category B with an action of t on it.
The category A/g(~,t)
of
these t-algebras admits a canonical enrichment to a V-category, which is B t.
In particular,
taking V = CAT, this applies to the case
K = 2-CAT. 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 2-cate$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 V-categories 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
2-categories
of Gray [ii]
so a finitely termi n a l
complete
object
~.4
2-category
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
2-cell
~: 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
t-algebra
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 monad-mapsT:
with
t-actions
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 2-CAT 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 2-monad)
for
in 2-CAT. For us,
D: K ~ K with,
then,
a doctrine
for its unit
on a 2-category
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
2-natural on the nose
have
the equalities
Z~berlein
[27]
calls
in (3.1),
The lax monads
D only
a lax 2-functor
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 2-functor
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 2-functor 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 "D-category" For m o r p h i s m s of D-algebras,
are the usual ones in nature,
and [17]
for "D-algebra".
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 D-algebras A,A', a
(f,f) where f: A ~ A' is an arrow in
K and ~ is a 2-cell 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
D-morphism
F strong
~ = i, so that
fn = n ' . D f .
of D - a l g e b r a s "
D-morphism
F : (f,l).
If we r e v e r s e and
F = (f,~)
(3.18),
"D-functor"
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
D-morphisms, write
(3.16),
also
we
call
an op-D-morphism.
D-morphism,
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
op-D-morphism. D-algebras of vertical with
the
pasting
same
We n o w m a k e For
and
of diagrams
objects
these
D-morphisms
~:
into
f ~
like
by restricting
D-morphisms
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
D-morphisms.
a D-2-cell
~:
F ~
G to
g satisfying
DA
Dc~
the
2-categories.
n (3 .i9)
under
~---
DA
n
~-A
b
Df'
f,
~
n'
A v
DA '
'~ A' n~
97
In the case
K = CAT we also
"D-2-cell".
With
D-morphisms,
and D-2-cells
write
as D-CAT
(here
the obvious
monadic
sub-2-category
it is this
of algebras)
that it is doctrinal,
or 2-monadic,
in 2-CAT, 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
D-morphisms
finite
that we m e a n
some
D-morphisms
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
2-cells.
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 D-CAT 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
D-Alg,
or as D-Cat
K = CAT
are considered;
for
laws of composltion, D-algebras,
by D - A I ~ the
2-category
transformation"
form a 2-category
in the case
We denote D-morphisms
say "D-natural
the
/23];
functors,
and
two m o n o i d a l
of @ over ~, with
K = CAT 2 the objects
of
D-Alg 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
D-Alg 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 2-cell.
not a 2-category: 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
D-Alg §3.4
= K D,
some
new
doctrine
sense: acted
questions
objects
of
K acted
D.
Write
U:
D-Alg
true
in the
o n by
we
formation
DU
functor
~Jvv~+ U.
G = UE:
sense
that
G: A ~
K with
D-Alg
have
about
ors A
~
the
as w e l l
matter
through
the
action
D-Alg
the
.
n and
2-functor
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
2-functors
op-lax;
the
when
certainly
to
laxity
the
in our
papers
K Then
is
not
corresponds
course
D-A/~
not
thing an
G only
to
that
a
to a in the
to
and
when we
A
the
do
object
Those 2-funct-
factorize is
to
of K b u t
definitions
ones
can,
there
of D on
right
given
as we
on U:
of
the
2-functor.
op-lax-natural.
A ~
A is
A ~
op-lax-natural-trans-
an a c t i o n G is
primary
2-functors
A ~ D-Alg
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
D-algebras
~ 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
2-functor.
arise
D-Alg
2-natural
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 ~
2-functor
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 d-Alg,
(3.20)
D'-Alg,
~
D-Alg,
to
Proposition
not
only
99
but also an evident
d-Alg s a t i s f y i n g
2-functor
d-Alg D'-Alg
(3.21)
~ D-Alg
K
where
of course
sense.
While,
the d-A/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 2-natural
d: D ~ D' that
P: D'-Alg ~ D-Alg
many more
op-lax-natural
d: D ~
Again we hope
to avoid
D'
d-Alg,
is 2-complete, 2-1imits
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
d-Alg 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
2-functor
d-Alg,
has a left adjoint.
category
there
axioms.
in suitable
It follows
functor
lying
map,
correspond
that
D' , and d-A/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
~ D-A/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 2-functor
,
- moreover
reasonable thought
in this
they
definit-
out the
situation;
j-Alg: D-Alg ~ l-Alg is just U: D-Alg ~ 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
6-6 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):
6-Alg
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 2-natural
commutative
d-,Alg (3.25)
K
transformation
101
in t h e
sense
that
U.~-Alg
Proposition
= identity.
3.5.
Any
We
2-natural
leave
transformation
D'-Alg~Jt~" ~
B-Algf o r It
there
are
a unique that
equivalent
doctrine
maps
d:
D'-A/g;
D-morphism
D'-Alg,
and
As ®:
A×A
each
D-algebra 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
D-Alg,
A•
with
A with,
for
isomorphisms
A D'-algebra
strict
D-Alg
equivalent•
category
a category
coherent
e°d ~ i
so t h a t
are
only
a monoidal be
that
2-categories
isomorphisms,
and
D-Alg
of
a D'-category
A n ~ A, a n d
are
maps.
in the
equivalent
D-Alg**
it
D,D',
~ D with
here
a D-algebra
let
m
d ~ d of d o c t r i n e
doctrines
sub-2-category
are
a D'-algebra
e.d
strong•
taken.
morphisms
whence
is
are
strict e:
to
D-Alg
6:
modifications
an e x a m p l e •
n ~ 0,
m
gives
is
~ A and
n I @ ( ® ,...,
the
(3.24)
D-morphisms
D ~ D'
of doctrine
since
D-Alg**
where
modification
follows
by modifications and
reader
d-Alg
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 D-algebra 0 1 n n-1 s e t t i n g @ = I, ® = i, @ = @(i• ~ ); again
strict
morphisms,
The
composite
D-Agg
~
the
composite
D'-Alg ~
so t ~ i s
comes
D'-Alg ~ D-Alg D-Agg
~
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, ~ D-Alg,
~ 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
70-122. 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
1-77.
Bifibration
in Math.
to bicategories,
in Math.
Cat6gories
80 (1963),
functor categories,
178-191.
in enriched
category
theory,
=
145 (1970).
structur6es,
Ann.
Sci. Ecole Norm.
Sup.
349-425.
[7]
C. E h r e s m a n n , Cat6$ories ' et structures
[8]
S. Eilenberg
and G.M. Kelly,
Categorical
Lecture
Al~ebra(La
(Dunod,
Paris,
Closed categories,
Jolla 1965.)
Proc.
(Springer,
1965). Conf.
on
New York,
1966). [9]
P. Gabriel and F. Ulmer,
Lokal pr~sentierbare
Lecture Notes in Math.
[ io]
J.W.
Gray, The categorical in Math.
[ 11]
J.W.
99 (1969),
Gray, Report in Z~rich, 195 (1971),
[ 12]
J.W.
=
221 (1971). comprehension
scheme,
Lecture Notes
242-312.
"The meeting of the Midwest Category Seminar
August
24-30,
1970",
Lecture Notes in Math.
248-255.
Gray, Formal category in Math.
Kategorien,
theory, to appear in Lecture Notes
103
[ 13]
A. Grothendieck,
Cat6gories
fibr6es et descente,
SSminaire
de
G@om@trie Al$@brique , Institut des Hautes Etudes Scientifiques,
[ 14]
G.M. Kelly,
Paris
Adjunction
in Math.
(1961).
for enriched categories,
106 (1969),
166-177.
[ 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
141-155.
categories
A triple miscellany:
doctrines,
and duality of functors,
315-349.
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
181-193. J. Pure and Applied
149-168.
Two constructions
Topologie
squares,
123-153.
of initial
R. Street, The formal theory of monads, Algebra,
[ 24]
195 (1971),
A characterization
of Algebra
of adjoint
et G@ometrie
on lax functors, Diff@rentielle
Cahiers de
XIII,
3 (1972),
217-264.
[ 25]
R. Street,
Fibrations
and Yoneda's
lemma in a 2-category,
(in this volume).
[ 26]
R. Street, Elementary
[ 27]
V. Z6berlein,
cosmoi,
(in this volume).
Doktrinen auf 2-Kategorien
Inst. der Univ.
ZUrich,
1973).
(Manuscript,
Math.
.FIBRATION$ AND YON.EDA'S ~EMMA IN A 2-CATEGORY by ~o~$ Street
Our purpose is to provide w i t h i n a 2-category a conceptual proof of a set-free 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 2-category 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 2-category is required to s a t i s f y an elementary completeness condition amounting to the existence of 2-pullbacks and comma objects.
This relates the
2-category closely to a 2-category 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 2-monad on the 2category of objects over B. Kock [ 4 ] .
This 2-monad is of a special kind distinguished by
We define lax algebras and lax homomorphisms for general 2-monads 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 2-monads.
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 2-category 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
category-valued 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 discrete-category-valued; that i s ,
105
set-valued.
With this interpretation of covering spans as set-valued 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 2-category. 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 2-category in @4. Note that, in general, for the 2-category V-Cat, 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 2-cell
U l
S~_~r/s
do
S
~- D
such that
f, = do@, q = d1@. In non-elementary terms,
r/s
is defined by a 2-natural 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 2-natural 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 2-pullbacks we say that
representable 2-category (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 2-category. / /
K are comma objects in
both representable and oprepresentable,
~
K°p.
In a 2-category which is
has a l e f t 2 - a d j o i n t
is automatically a 2-1imit in
#
and any l i m i t
which e x i s t s in
K0
Proposition i.
In a representable 2-category each opspan has a comma object.
The formula is
Proof.
K.
r / s : s*otDor.
// In a representable 2-category, an i d e n t i t y 2-cell
arrow
i:A
~ tA,
I A ~ I~A I
corresponds to an
and the composite 2-cell ~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 2-category 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 2-cell
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 well-known 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
2-functor.//
§2. Lax algebras and f i b r a t i o n s Suppose
D is a 2-monad on a 2-category
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 D-algebra 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 2-category 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 D-algebra is a lax D-algebra in which
are equal as indicated.
A normalized lax D-algebra has
isomorphisms.
D-algebra is a lax D-algebra with both in
C,
DE with
Kock [ 4] c -~iD
CE:D2E
identities.
~ DE is the free D-algebra on
are
A
Of course, for any E E.
has distinguished those 2-monads D with the property that
in the 2-functor 2-category 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 D-algebra 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 2-monad with the Kock property and suppose
the 2-cell 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 2-cell
D 2 E C ~0 ~ E
iDE-i E 2-cell
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 2-cell
Proof.
(a)
gives
0
E,C,~
enriches
Let
with the structure of pseudo D-algebra.
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 D-algebras 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 2-monad with the Kock property and suppose
Suppose
is an arrow between pseudo D-algebras.
l:Df.iE----~ i E , . f
which corresponds under adjunction to the identity 2-cell unique with the property that equality (5) holds, f
ef:c. Df
Then the 2-cell
Furthermore, this
~ fc is
enriches
9f
with the structure of lax homomorphism.
Suppose 9f
Proof.
is as explained in the proposition.
since both the 2-cells iE,f
1 ~ iE,f
ciE,f
~ f
correspond to the i d e n t i t y 2-cell
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 2-cells i d e n t i t y 2-cell 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 2-category 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 2-cell
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 2-cells
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 2-cell
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 2-pullback 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 2-cell loi
~B ~
c ~, L
L(E,p) = (tBop,dlp , Lf = 1of, L~ = io~.
denote the 2-natural transformations w i t h iol E
The diagrams which say t h a t say t h a t
L
is a 2-monad 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 O-fibration over
the s t r u c t u r e of pseudo L-algebra.
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 L-algebra. Proposition 9. over
B
(Chevalley criterion].
if and only if the arrow
The arrow
p : ~ E ....~ p / B
p:E
> B is a O-fibration
corresponding
to the 2-cell
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 L-algebra.
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
l-q
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 O-fibration will
be c a l l e d
normal when there is a normalized pseudo L-algebra 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 2-categories 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 O-fibra~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
2-natural
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 2-category 2-CAT. g:B' ........> B
Suppose
K.
is an arrow in
For each lax L-algebra
E, the arrow V
gE
Lg*E enriches
f:E
g*E
~ E'
g*(c)
~ g*LE
g*E
>
with the structure of lax L-algebra.
For each lax homomorphism
of lax L-algebras, the 2-cell 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 L-algebra or an L-algebra 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) O-fibration along any arrow is a (split)
O-fibration.//
Let
R:KA
~ KA denote the 2-functor given by:
f (E,q)ZIC(E'
A/f ,q')
'
(A/q'd0)
~
g
A/g
There is a 2-monad structure on K by
Kc°.
An arrow
(A/q',d0).
q:E
R and the theory develops as for
>A
is called a 1-fibration over
A
L;
j u s t replace
when (E,q)
supports the structure of pseudo R-algebra. Note that the category
SPN(A,B)
of spans from
2-category 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 2-monad 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 M-algebra.
A
to
B when i t
A split bifibration is an M-algebra.
Results on L-algebras and R-algebras can be t r a n s f e r r e d to M-algebras 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 M-algebra structures
are in bijective correspondence with pairs of arrows of spans
~ E,
structures on
E
CR:EotA E
, E
c L, c R are L-algebra, R-algebra
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 M-algebras if and only if it is a homomorphism of both the corresponding L-algebras and the corresponding R-algebras.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 2-category. Again we work in a representable 2-category
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 M-algebra 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 L-algebra and R-algebra
By d u a l i t y , i t suffices to show that
f
commutes with
j u s t the L-algebra structures. The L-algebra 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 M-algebra 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 2-cells
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 2-cell
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 2-cell u
there exists a unique 2-cell
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 2-cell
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 2-cel 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
2-cell
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 O-fibration.
extension of
f
p
along
The 2-cell
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 2-cell
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 2-cell 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 2-cell 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 2-cell 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) 1-173.
[2]
J.W. Gray, Fibred and cofibred categories. Proc. Conference on Categorical Alg. at La Jolla (Springer, 1966) 21-83.
[3]
J.W. Gray, Report on the meeting of the Midwest Category Seminar in Zurich. Lecture Notes in Math. 195 (1971) 248-255.
[4]
A. Kock, Monads f o r which structures are a ~ ' o i n t to ~ i t s .
Aarhus University
Preprint Series 35 (1972-73) 1-15. [5]
F.W. Lawvere, The category of categories as a foundation for mathematics. Proc. Conference on Categorical Algebra at La Jolla (Springer, 1966) 1-20.
[6]
R.H. Street, The formal theory o f monads. Journal of Pure and Applied Algebra 2 (1972) 149-168.
[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) 217-264.
ELEMENTARY COSMOI I by Ross S~re~,t
The theory of categories enriched in some base closed category V, is couched in set-theory; 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 2-category
V-Cat of small V-enriched categories together with the 2-functor P:(V-Cc~t)o°°P÷V-Cat given by PA = lAMP,V] (= the V-enriched category of V-functors 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 pseudo-natural equivalence between the category of functors from B to PA and the category of covering spans from A to B.
Generalizing to a representable 2-category
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 2-functor.)
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 V-valued V-enriched 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 well-known one in the theory of V-enriched categories concerning the relationship between comma objects and pointwise kan extensions.
I f we naively take K to be the
2-category V-C~ 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 V-Co~twhich
is an elementary cosmos and provides f u l l e r information on V-enriched 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 set-valued presheaf construction (compare Lawvere [14]). Also the (pre-) ordered objects in any elementary topos provide an example of an elementary cosmos; in particular, the 2-category of ordered-set-valued 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 ever-present ordered objects in a topos.
The
2-category of category-valued 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 CoJt-enriched categories ( t h a t i s , 2-categories) 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; 2-categories 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 ; hom-statement of the Yoneda lemma; extension and l i f t i n g
properties
of hom-arrows.
§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 Chevalley-Beck 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 2-categories; 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 eilenberg-moore 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 pre-Spanier construction; categories.
138 1.
Internal a t t r i b u t e s . Let
category
K be a representable 2-category. 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 M-algebras 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 M-algebra.
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 o-of: 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*o-of: 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 2-functor 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 J-g 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 2-cell 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 2-cell 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 ;
1-fibrations have been considered by
A O-fibration over a category
B corresponds to a pseudo functor
when the original O-fibration is also a 1 - f i b r a t i o n , the functor
has a r i g h t adjoint
and Benabou-Roubaud [ 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 2-cell
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 O-fibration,
b: G
~ B is an arrow,
Then the 2-cell
PE
)" PEb
¥p
p
PB
~
PG
Pb
corresponding under adjunction to the identity 2-cell 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 set-valued 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 2-cell 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 2-cell
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 2-cell
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 2-cell ~,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:
2-cell:
is the isomorphism of Theorem 8.#
and each legitimate object
K(K,A)
lifts to a 2-functor
L
Given arrows
f: A
~ X,
A
of
K, the
with the structure of a VK-category,
with its closed category structure.
Indeed, the 2-functor
........>' 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
2-cells: 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
G-cocomplete 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 :
G-cocomplete 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
G-cocomplete then
of
such that
PG
is G-cocomplete
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 VK-category. ~
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)
sub-2-category
denote the pseudo functor which is the i d e n t i t y
on objects and which is given on hom-categories 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 2-cell
~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 non-symmetric 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 2-cells
a
and 2-cells
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 non-uniform case) we require
K to be a cosmos.
A cosmos is a precosmos for which the 2-functor
adjoint
P*: K
P: Kc ° ° p
>K
has a l e f t 2-
This means of course that there is a 2-natural 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 lax-fibre 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 lax-fibre 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 2-adjunction
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 2-CAT 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
2-CAT, 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 2-functors out of
~
with arrows F:
A---~K.
If
with objects of the target 2-category, 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 2-functors
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 2-enriched; 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 2-functor
in the 2-category
> Co~t is a monad in
K Oxt.
Take A to be the 2-category 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 2-functor
We take this monad @ to be the monad called
AA by Lawvere in [15] pp150-1. 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 2-category 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 2-cell 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 2-category.~
168
Theorem 33.
A representaSle 2-category with a 2-terminal object admits the follow-
ing constructions: (a) finite limits; (b) the construction of algebras; (c) oplocalization. These constructions are all 2-enriched. 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 2-cell 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 2-fun~tors such that 9 ® F
> 11 is a 2-functor with a right 2-adjoint 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 eilenberg-moore 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
eilenberg-moore 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 co-eilenberg-moore 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 co-eilenberg-moore object for the comonad
that is: PK ,,
1
)-PK
PA
>PA Ps
is the universal Theorem 35. that
(A,s)
P: Kc°°p
Ps-coalgebra.
Suppose >K
K
is a representable 2-category endowed with attributes such
has a left 2-adjoint.
Suppose
is a monad for which the kleisli object
the eilenberg-moore 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 Ps-coalgebra, 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 s-algebras 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
eilenberg-moore 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)
eilenberg-moore 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 Lawvere-Tierney [ 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
"power-object" 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 sub-2-category 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 power-object
is a representable 2-category with f i n i t e 2-1imits. 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 2-category with f i n i t e 2-1imits.
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 2-functor
P: Kc°°P--*K.
Pdl, Pdo:PA0 --~PAI.
inc PA
~ PA0
inc I
~
PAO
IPdo ~ PA1
.
Pd1 The 2-functor 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 RP-Z(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 sub-pseudo-functor 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 2-categoz~
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 2-adjoint
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 order-preserving arrow
f : A --+B, the order-preserving arrow
Pf: PB --*FA has both a left and a right a~oint.#
2)
The pre-Sp~nier constuction as a
P.
The 2-category Simp and 2-functor
P presented here are taken from unpublished
work of Day-Kelly on categories l i k e categories of topological spaces. A function
f : X--+Y
between sets
through the one-point 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 2-category whose objects are simple categories, whose arrows
are simple functors, and whose 2-cells 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 2-category. 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' I-T-~_, 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 2-functor 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 2-cells 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
2-functors
x CATfT ~ CAT ,
(B,A)) to A and B respectively.
clear from (4.12) Proposition
~ AB k
2-functors
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 2-natural transformations
do: J ~ forg O and d1: # ~ forg I. D
5. 5.1
THE 2-FUNCTOR
o AND THE EMBEDDING
Form the composite
CATf*T x
CAT~*CAT ~ [CAT,CAT]
2-functor
T---~----~
CAT[*T
×
CATfT
i × r • where r m
(I,~B~),
: Y ~ CAT~T etc.
is the 2-functor
Our interest
write the above 2-functor (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 2-cells 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
3-category
2-functors
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
2-functors the
closed;
K × L ~ M and
2-functor
o of
(5.2)
a 2-functor
(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:
2-functor
a 2-natural 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 2-natural 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 2-natural-
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 2-cells.
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 A-FA' are indeed
the naturality
of
D
2-catesories.
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 ,
2-functor
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 category-with-coequalizers. 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,
2-categories.
So for a 2 - f u n c t o r
~ I and
image
D:
~ [CAT,CAT]of
is not
category.
(l,rn~)oB
2-functor
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 2-natural
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)
2-naturally (5.25)
(DB,B!)
if the i n d i c a t e d
a right
2-reflexion
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 1-cells,
but always
a sub-2-category taken
to be full
208
on 2-cells. 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 2-cells,
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
2-functor
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 n-ad
augmentation
category
of
(that
showing
derived
of 51.2 above