Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, ZiJrich
391 John W. Gray University of Illi...
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Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, ZiJrich
391 John W. Gray University of Illinois at Urbana-Champaign, Urbana, II/USA
Formal Category Theory: Adjointness for 2-Categories
Springer-Verlag Berlin.Heidelberg • New York 1974
AMS Subject Classifications (1970): Primary: 18D05, 18D25 Secondary: 18A25, 1 8 A 4 0
ISBN 3-540-06830-9 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-38?-06830-9 Springer-Verlag New Y o r k . 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. Library of Congress Catalog Card Number 74-7910. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Contents
Introduction
I,i
Cate@0ries
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Yoneda
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Adjointness
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Fibrations
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1,2
2-Categories 2-functors
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Cat-natural
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12
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13
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14
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16
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25
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3-comma c a t e g o r y double c a t e g o r y
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2-and 3 - c a t e g o r i c a l .
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28
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31
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32
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33
fibrations . . . . . . . . . . . .
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1 11
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2-comma c a t e g o r y
Bicategor ies
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3-category
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transformations
Modifications
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transformations
Quasi-natural
1,3
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A d j o i n t Functor T h e o r e m Kan extensions
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35
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38
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4O
Quasi-natural transformations
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43
Examples
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Pseudo-functor s .
Bim
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Spans
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%
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Bim
(B) .
Bim
(Spans
Fibrations
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45
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46
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48
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5O
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IV 1,4
Properties
of Fun(A,B)
Quasi-functor
Characterization Composition
and Pseud
of two variables of Fun(A,B)
quasi-functor
Quasi-functor
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Quasid-natural Quasi -functor
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x
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Appendix ~.Universal A p p e n d i x B. Iso-Fun
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69
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enriched
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86
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Explicit
formulas over
Fibration
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V 1
×
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V 2 .
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101
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111
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115
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120
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Uniqueness,
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in 2-categories
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composition
Adjoint Squares
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124
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134
136 ~37
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and preservation . . . . .
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106
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properties . . . . . . . . . . . . . . . .
Adjoint morphisms .
103
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and monoid properties
Homomorphism
96
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92 95
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81 83
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in 2-Cat®
73 80
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59 67
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structure . . . . . . . . .
copseudo-functor
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Functors
Examples
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property . . . . . . . . . . . . . . . . . .
Composition
Examples
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Pr0.P.erties of 2-comma categories Universal
1,6
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Appendix ~.Categories 1,5
56
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transformations . . . . . . . . . . . .
Monoidal closed category (A,B)
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55
of n-variables . . . . . . . . . . . . .
Tensor product
Pseud
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(A,B) . . . . . . .
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139
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144
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152
Kan extensions
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154
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156
Examples
Examples
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Formal criterion
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for adjoint .
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158
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160
Interchange of limits
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161
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Cocompleteness
Final
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V
1,7
Quasi-adjointness Definitions
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Uniqueness,
composition
Transcendental Universal Examples
mapping
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References Bibliography
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in C a t .
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Comprehension
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217
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Scheme ......
244 251
Morphisms .........
265
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272
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224 237
Lemma ...............
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201
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Adjunction .
. . . . .
197
The Quasi-Yoneda
.
~
in C a t . . . . . . . . . . . . . . .
The Categorical
of Symbols
180 187
quasi-limits ..............
extensions
Globalized
177
187
Quasi-fibrations Quasi-Kan
169
principles ...............
Quasi-limits
.
. . . . . .
............
. . . . . . . . . . . . . . . . . . . . . .
Some Finite
168
properties ..............
Quasi-colimits
Index
and p r e s e r v a t i o n
quasi-adjunction
Some general
Table
166
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275
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279
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28O
Introduction
The purpose of c a t e g o r y t h e o r y is to try to d e s c r i b e c e r t a i n general aspects of the structure of m a t h e matics.
Since c a t e g o r y t h e o r y is also part of mathematics,
this
c a t e g o r i c a l type of d e s c r i p t i o n should a p p l y to it as well as to other parts of mathematics.
When I first c o n d u c t e d a seminar
on this subject during the Bowdoin Summer Session on C a t e g o r y T h e o r y in 1969, Saunders Mac Lane suggested the name "Formal C a t e g o r y Theory"
for this study. The basic
c a t e g o r y of small categories, Cat,
idea is that the
is a 2 - c a t e g o r y with pro-
perties and that one should attempt to identify those properties that enable one to do the "structural parts of c a t e g o r y theory". The results of this present
study suggest the following a n a l o g y
w i t h h o m o l o g i c a l algebra: Cat abelian groups~ a category modules~
%
c o r r e s p o n d s to the c a t e g o r y of
the categories,
Cat x , of c a t e g o r y objects
in
w i t h p u l l b a c k s c o r r e s p o n d to categories of
and r e p r e s e n t a b l e
2-categories c o r r e s p o n d to abelian
categories. There has been a c o n s i d e r a b l e a m o u n t
of w o r k on various
aspects of this study. Much of the w o r k of E h r e s m a n n is d i r e c t l y c o n c e r n e d with, or at least, relevant to it. In particular, he has studied see, e.g.,
Cat % [~2],
both published,
under the name of " p - s t r u c t u r e d categories",
[13], e.g.,
[i4], [BC],
relevant. M a r a n d a ' s paper, in its first part,
[15 ]. Similarly,
Benabou's work,
[ 3 ], and unpublished, "Formal C a t e g o r i e s "
[36 ]
[ 4 ] is contains
a d i s c u s s i o n of adjoint squares very similar
to I,§6. However, by a formal category, he m e a n s a c a t e g o r y
VIII over a m o n o i d a l c a t e g o r y in the sense of [E-K]. The papers of Bunge
[7 ] and P a l m q u i s t
[37], are in the same spirit,
as is recent u n p u b l i s h e d w o r k of Street.
In general, much of
the w o r k in closed c a t e g o r i e s needs o n l y to be r e p h r a s e d to apply to 2-categories. However,
Some of this is b r i e f l y done in I,§6.
the m a i n thrust of this w o r k does not seem to have
been noticed before; namely,
that complete,
2-categories are q u a s i - c o m p l e t e what
(III,§4)
is needed to do c a t e g o r y t h e o r y This volume
representable
and that this is
(I, §i and IV).
is the first part of a projected three part
w o r k and its sections are numbered
I,l through 1,7.
In it
are to be found general background m a t e r i a l and the various r e l e v a n t notions of adjointness.
In I,§l the notations for
c a t e g o r i e s are fixed and the v a r i o u s aspects of the c a t e g o r y , w h i c h are r e s p o n s i b l e for the p r o p e r t i e s of w i s h to abstract, statement
are discussed,
Cat
we
leading up to a precise
(in I,I.9 through I,I.13)
of those "structural parts
of c a t e g o r y theory" w h i c h are amenable to the sort of treatm e n t presented here. If one tries to formulate standard results about categories as global
statements about the 2 - c a t e g o r y
Cat
order to try to discuss them in other 2-categories,
in
then it
f r e q u e n t l y h a p p e n s that this involves "functors" that m a y not r e a l l y be functors but o n l y p s e u d o - f u n c t o r s and w h i c h look like adjoints,
except that there are extra natural t r a n s f o r m a -
tions expressing various kinds of global compatibility.
Thus,
it is not sufficient to deal solely with 2 - c a t e g o r i e s and 2-functors. There are,
in fact,
four types of s t r u c t u r e s w h i c h
arise in trying to discuss these situations:
IX a) 2-categories.
(I,2.1 - 1,2.3)
b) The c a t e g o r y of 2-categories with the closed structure given by the 2-category tions, denoted by
Fun(A,~)
transforma-
2-Cat O . (I,2.4 and 1,4)
c) Bicategories. d) The partial to b).
of quasi-natural
(I,3)
structure
for bicategories
corresponding
(I,4.21) Frequently,
the most difficult
which pattern a given adjointness for instance,
appear as l-cells
in the 2-category
e) 2-comma categories.
g) Double
quasi-natural
transformations
Fun(~,~
in describing
and as 2-cells
their properties.
(I,2.5)
and 3-comma categories.
and triple categories.
Because of this profusion of structures, be regarded as encyclopedias However,
fits. One reason
in b). Besides these structures,
there are others which enter
f) 3-categories
is to recognize
situation
for this is that,
in the situation described
thing
(I,2.6| 1,2.7)
(I,2.8) 1,2 and 1,3 should
to be refered to as needed.
scattered through them are a number of examples w h i c h
might be helpful
to the reader.
the important b i c a t e g o r y of assertions
Bim(Spans
for which no specific
later in this paper or elsewhere) of Benabou,
Ehresmann,
bibliography.
In particular,
Gray,
X)
in 1,3.4
(3)
is described. Proofs
reference
is given
(either
can be found in the works
and Lawvere cited
in the
X The m a t e r i a l the c o n s t r u c t i o n category built internal nation
homs
proper
of the n o n - s y m m e t r i c a l
on the c a t e g o r y is given by
of the r e l a t i o n s
constructions an e x p l i c i t
one
In I,§6,
assertion
g o r y are discussed. Kan extensions, This
Using
adj o i n t s
where
these
thereby provide discussed found
the
with
formal
a number
at the b e g i n n i n g
properties
situation
sections
the Bowdoin mately
was
in the
Part Cat
%
II will
(II,§i)
category
object
formal
bicategory
be devoted
Bim(Spans
properties.
in I,§7 on quasi-
Many examples
(II,§2). i)
above,
in a series
fur M a t h e m a t i k
%
and
fail d r a m a t i c a l l y ,
in a m u c h
Session mentioned
and are
of these
cruder and
fashion
later,
of lectures
last at
approxiat the
of the ETH in ZUrich.
to the
and the c a t e g o r i e s in
squares
of 1,7.
form g i v e n here,
Forschungsinstitut
of 2-comma
section. A list of them will be
first p r e s e n t e d
Summer
this
in a 2-cate-
of adjoint
Much of the c o n t e n t four
inspired
of q u a s i - a d j o i n t s .
i-cells
of new phenomena.
at the end of this
(I,4.9) , and,
the e x p e c t e d
properties
Two
[ 4 ]. In I,§5,
of adjoint
the
homs.
fibred c a t e g o r i e s ,
language
exami-
of 2-categories,
and r e l a t i o n s
to the d i s c u s s i o n
one e a s i l y derives
is to be c o n t r a s t e d
product
the algebraic
the p r o p e r t i e s
one of w h o s e
internal
of Benabou,
are c e n t r a l
closed
, and the d e t a i l e d
for the tensor
to derive
which
monoidal
the two
(I,4.23) , one using
is used
categories
Fun(X,~)
in terms of cells
by a more g e n e r a l
in I,§4 w i t h
of 2-categories,
between
are g i v e n
in an a p p e n d i x
material
to this w o r k b e g i n s
study of the c a t e g o r i e s
of ~-valued
These
functors
r e a l l y make
and we provide
a Yoneda
on a
up the embedding
XI into this bicategory.
In II,§3, we study the r e l a t i o n s of
these notions to fibrations
in order to have the results
-
via the embedding theorem b e l o w - for s t r o n g l y r e p r e s e n t a b l e 2-categories.
Finally,
in II,§4, we treat the important
special case of c a t e g o r y objects
in the c a t e g o r y of triples.
Part III will be c o n c e r n e d w i t h r e p r e s e n t a b l e 2-categories (III,§!). The term was suggested by Jon Beck since they are c h a r a c t e r i z e d by the p r o p e r t y that the 2-cells are "representable" by l-cells.
In III,§2, we give a c l a s s i f i c a t i o n theorem
in terms of c a t e g o r y objects
in the c a t e g o r y of triples. These
2-categories w e r e d i s c u s s e d from this point of view at Bowdoin. From the present point of view, a r e p r e s e n t a b l e
2 - c a t e g o r y is
a 2 - c a t e g o r y w i t h a suitable p r o p e r t y rather than a c a t e g o r y with a d d i t i o n a l
structure.
In III,§3,
2-categories are introduced. into 2-categories of the form
T h e y admit
"strict" embeddings
Cat ~ . In III,§4, c o m p l e t e n e s s
theorems are g i v e n for r e p r e s e n t a b l e c o m p l e t e n e s s hypotheses,
strongly representable
2-categories.
these share w i t h
Cat
Under o r d i n a r y
the p r o p e r t y
of admitting all cartesian quasi-limits, while corepresentable, cocomplete
2-categories are c a r t e s i a n quasi-eocomplete.
in particular,
using the results of Street,
Thus,
these 2-categories
admit c o n s t r u c t i o n s of Kleisli c a t e g o r i e s and E i l e n b e r g - M o o r e categories
for triples
(cf. 1,7) . III,§5 is devoted to m a n y
e x a m p l e s of r e p r e s e n t a b l e and c o r e p r e s e n t a b l e
2-categories as
well as examples of w e a k e r types of structures. We m e n t i o n p a r t i c u l a r that
Cat %
w i t h strong pullbacks) the algebraic
and
~-Cat
in
(for a closed m o n o i d a l
are s t r o n g l y representable.
In III,§6,
structure of comma objects n e c e s s a r y for c a t e g o r y
XII theory
is discussed. Part
context
IV will
study the s t r u c t u r a l
of r e p r e s e n t a b l e
2-categories,
theorems
of I, in the
in so far as this
is
possible. In an A p p e n d i x this
for
foundations,
of a r e p r e s e n t a b l e in C h a p t e r s out
2-categories.
This
like the adjoint
This
the
the n o t i o n
serves
functor
complete
Zurich,
the
as well
Summer
express
m y appreciation.
I.
those p a r t s
sy n t a c t i c a l
long term
as the
Geneva,
support
I would
for m a k i n g
impetus
above, also
possible
(which have
and h e n c e
considerations.
and s e l f - c o n t a i n e d
Session m e n t i o n e d
of things
and Kan e x t e n s i o n s
and the F o r s c h u n g s i n s t i t u t
Bowdoin
of Part
t h e o r y of r e p r e s e n t a b l e
to identify theorem
as is the m a t e r i a l
l e n g t h y work w o u l d n e v e r h a v e r e a c h e d
Foundation
Institute,
since
of
t h e o r y can be c a r r i e d
of s e t - t h e o r e t i c a l
the m a i n r e s u l t s w i t h o u t
of the ETH,
of c a t e g o r y
that are p u r e l y
independent
that,
is e l e m e n t a r y
of the e l e m e n t a r y
form of a r e a s o n a b l e
Science
idea being
2-category
logicians)
completely
implications
III and IV, m u c h
in the context
concerned
to Part IV, we shall!/ / comment b r i e f l y on the
its p r e s e n t treatment
of
of the N a t i o n a l fur M a t h e m a t i k
provided
b y the
for all of w h i c h
I
like to t h a n k the B a t t e l l e the e a r l y a p p e a r a n c e
I,i
Part
I: A d j o i n t n e s s
for 2 - c a t e g o r i e s
I,i. C a t e q o r i e s . category tively,
of small a model
In this work,
sets
in some
of the t h e o r y
Lawvere-Tierney
[22].)
small hom
sets.
A category
is small,
The c a t e g o r y
by
Cat
. When
categories
belonging
that we
in C h a p t e r
Cat
categories
(see
"
I,I.2°
[E-K])
~
inclusion
phisms) (path)
x -
to have
will
Cat
~Cat
be d e n o t e d for
big enough
(large)
to c o n t a i n
has m a n y p r o p e r t i e s
of r e p r e s e n t a b l e
closed;
not be
2-categories
for all
--I
components" in either
o
functor,
~
Let (i.e.,
: Cat ~ Sets where
direction;
,
to".)
set functor
of Sets as d i s c r e t e ~
exponentiation
(_)A
of adjunctions.
Let
i.e.,
a closed category
"is left adjoint
The u n d e r l y i n g
categories.
morphisms
sense of
are a s s u m e d
categories
describes
such that
I " means
is part of a string the
(or a l t e r n a -
use of but w h i c h w i l l
is c a r t e s i a n
A
(The sign
the
III.
functor
structure
denote
if its set of m o r p h i s m s
to some u n i v e r s e
frequent
in the
shall w r i t e
in our d i s c u s s i o n
I,l.l. via
of small
constructions.
shall make
incorporated
of sets
is small
we
Sets will
fixed u n i v e r s e
All c a t e g o r i e s
necessary,
all the r e l e v a n t
i
a path
e.g.
I (-)I D
: Cat ~ Sets
: Sets ~ Cat
only
be
i d e n t i t y mor-
be the
"set of
is a string
of
I,l
Let
G : Sets ~ Cat
functor~
i.e.,
if
2
be the "trivial c o n n e c t e d groupoid" X
is a set then
G(X)
is the g r o u p o i d
(= c a t e g o r y w i t h every m o r p h i s m an isomorphism) X
and such that there is e x a c t l y one m o r p h i s m
y
for all
x
and
y
J l(-)I
,G
x
to
.
is c o m p l e t e and cocomplete.
limits and c o l i m i t s are Cat-limits, categories~
from
X . Then
~D
o I,I.3. Cat
in
with objects
Furthermore,
in the sense of closed
i.e.,
~ l ! m ~ i = !im(B_A--i) ; (lim Bi )A = l!m(BiA)
This of
follows i m m e d i a t e l y from the c a r t e s i a n c l o s e d structure Cat
. Limits are given by the usual subobjects of prod-
ucts. The structure of colimits
is more complicated.
Coprod-
ucts are d i s j o i n t unions, while a c o e q u a l i z e r F G is d e s c r i b e d as follows: is the c o e q u a l i z e r of ~(Q,Q')
I (-) I
IFI
has a right adjoint so
and
IGI
121
in Sets. The hom set
is a c o e q u a l i z e r of the c o p r o d u c t of all finite
p r o d u c t s of the form
a(BI,B l) × a(Bz,B ~) × ... × ~(Bk,B {) where
P(B i) = Q
, P(B~)
= P ( B i + i)
then an a d d i t i o n c o p r o d u c t w i t h ~ whose c o e q u a l i z e r maps
, P(B{)
= Q'
. If
Q = Q'
is taken. The two maps
is formed are c o n s t r u c t e d b y inserting the
,
I,l
3
FA, A, j
B(FA,FA')
~
B(GA,GA')
/I A(A,A')
i ~
B(B,B') B_(B',B")
°~6~
B (B,B,,)
~
B(B,B)
~(B,B')xB(B',B")
I
into all possible coproduct. natively,
positions
For an explicit the morphisms
[29] as equivalence phisms
in all possible products formula,
see Wolff
can be described
in the
[40]. Alter-
as in Lawvere
classes of admissable
strings of mor-
from
fl,---,fn where a string
is admissable
valent to the codomain valent
of
fi+l
fi
is equi-
, and two strings are equi-
if they are made so by the smallest equivalence
lation compatible
with c o m p o s i t i o n
are equivalent whenever G(q)
if the domain of
are e q u i v a l e n t
f'f
category
~
defined
for all morphisms
1,1.4. The properties to representable
of
2-categories
. Here, ~
identity morphism;
~
such that
denotes denotes
Cat
in
f',f
B , and g ~~
re-
and
f'f
F(g)
and
.
w h i c h will be extended
all depend u l t i m a t e l y on the the c a t e g o r y with a single the c a t e g o r y that looks like
I,i
0 ~ 1 , with i = 0,1
~ 3-
Oi : ~ ~ ~
the functors given by
O1
2
to
y : ~ ~ 33-
Oi(A)
= i ,
denotes the p u s h o u t
1
and
4
(i.e., ~
~
->
2
~
3_
the o n l y other n o n - t r i v i a l
functor from
looks like
)
1
)~
2
and
~
denotes the pushout
2
~
3
p
--
~
3
.>
4
The higher ordinals can be d e f i n e d similarly,
or by induc-
tion as the p u s h o u t of
n ~
~}
Oo
81
!
*2
w i l l denote the c a t e g o r y w h i c h
c a t e g o r y of
Cat
d e t e r m i n e d by
[ccFM]). A l t e r n a t i v e l y , the o r d i n a l s
1,2,3
and
looks like the full sub-
!,~,3-
and
~
(cf. Lawvere
it is the c a t e g o r y w h e r e objects are 4
and w h o s e m o r p h i s m s are all
I,l
5
order p r e s e r v f n g maps b e t w e e n them. As such it is a full s u b c a t e g o r y of the c a t e g o r y
~
of all finite ordinals
(including O) and all order p r e s e r v i n g
functions.
1,1.5. The entire t h e o r y p r e s e n t e d in Part III of this w o r k is a r e f l e c t i o n of the r e l a t i o n s b e t w e e n the categories
!,~,~,~
, and
~x~
. Of these, ! , ~ , ~
d e t e r m i n e the e l e m e n t a r y structure of categories, sense that, for instance, Cat of limit p r e s e r v i n g [CCFM].)
functors
!
in the
is isomorphic to the c a t e g o r y from
{~}op
to Sets
(cf.
It is the thesis of the study of r e p r e s e n t a b l e
2-categories that, by including tion by these five categories, g o r y theory tween
, and
!,~
~×~
as w e l l as e x p o n e n t i a -
one r e c a p t u r e s m u c h of cate-
First of all, the n o n - t r i v i a l relations beand
~x~
can be s u m m a r i z e d by the statement
that the functor
- x 2 : Cat ~ Cat
is a c o t r i p l e A : ~ ~ ~x~
(see 1,7) w i t h c o m u l t i p l i c a t i o n the d i a g o n a l and counit the c o n s t a n t
By adjointness,
(-)~
(_)4
(_)T
and unit Now,
functor
T : ~ ~ !
•
is a triple w i t h m u l t i p l i c a t i o n
small c a t e g o r i e s are the objects of another
category, Cat t , whose m o r p h i s m s are natural transformations. These can be identified either w i t h functors with
functors
~x~ ~ ~
~ ~ B~2 | i.e., as c o K l e i s l i m o r p h i s m s
the c o t r i p l e or as Kleisli m o r p h i s m s
or for
for the triple. C o m p o -
sition is g i v e n by K l e i s l i composition,
so, for instance,
1,1
Cat t
is isomorphic
((_)~
(_)A
(_) T)
to the Kleisli
in
Cat
for the total c a t e g o r y given
in III,
. A similar
Cat t
taken by a n a t u r a l
of the triple
representation
"set of m o r p h i s m s "
I (-)~i
on
category
of a r e p r e s e n t a b l e
§2. The
is also d e f i n e d
6
theorem
2-category
is
functor
: Cat ~ Sets
, a morphism
transformation
e
f : 2 ~ A : ~x~ ~ ~
being to the mor-
phism
•f
Associativity functor.
=
~(f)
Z
:
of Kleisli
As such,
~>ZxZ f - ~ composition
A×Z shows
it is right adjoint
to
~ that
D
this
is a
: Sets ~ Cat t
i.e.,
D
I,I.6. ture of the
,
The next
functors
1(-)21 thing
between
(on Catt). to observe ~,~,~
and
is that the ~
struc-
implies
that
~o 1
constitutes hence
(-)~
a cocategory
81
object
is a c a t e g o r y
functors
on
Cat
functors
for any
~ i.e., ~
£ Cat
Y
2
in
object
Cat
(see II,
in the c a t e g o r y
there are a p u l l b a c k
§I)
and
of endo-
diagram
and
I,l
7
Pr2--C~
>
c2
prcC--
d =C o --
c_ 2
d
satisfying
the e q u a t i o n s
1,1.7.
It t u r n s
Parts
III and
ness,
etc.,
functors
-
and natural there
indicated
1
~
are
the Y o n e d a
structure
functors in
~×~
object.
in the t h i n g s
treated lemma,
is p r o v i d e d
transformations
triangles
>
that
fibrations,
- the c r u c i a l
Explicitely, the
IV
c!
l=C 51
for a c a t e g o r y
out
o
u
between and
adjoint-
b y the
~
[
in
and
taking
~×~ ~
.
to
:
(1,1)
(o,i)
2
f (o,o)
0
Or,
treating
out
and denoting
{...}
a
: ~ ~ ~ maps
as the
into
first
limits
-
-
(1,o)
~
injection
and out
in the p u s h -
of c o l i m i t s
by
, one c a n w r i t e
u
=
{{Co, x} , {X,~I}}
= {(~,~o } , {~i,X}} where (i.e., and
~. : O,T l l not
they
: ~ ~ ~
factoring
satisfy
. These
fihrough
2)
the e q u a t i o n s
are
the o n l y n o n - t r i v i a l
functors
from
~
to
~×~
1,1
u¥
By e x p o n e n t i a t i o n ,
this
=
e¥
says
8
=A
that
.
for e v e r y
~
~ Cat
, the
diagram
2. 2 2×2 C--) ---~C_----
LC =c--u
C3
~_>
--
commutes. (-)~ Cat
C2
C
We s h a l l
being
C3
>
see
in II,
§4 that
this
is e q u i v a l e n t
a category
object
in the c a t e g o r y
of t r i p l e s
to on
.
1,1.8.
It is of c e n t r a l
importance
2
that
->
the
square
3
I 3
is a p u s h o u t
'>
(and h e n c e
is a p u l l b a c k ) .
This
functors
from
"folding
functor",
as
follows
2×2
(here
= {3,y{/,
for a n y
allows to
fd = u
3
one
{3,3}
G = {3,¥{ao,2}} = {y~,q},3_} ,4
= {~{%,_2},3}
, the p r e c e e d i n g
to d e s c r i b e
. There
is t r e a t e d
%}}
C
2x2
and
are
five
four
as the
square
the n o n - t r i v i a l such.
others, first
The described
injection):
I,l
In terms
of pictures, while
9
u
is t h e
identity
on
u
and
takes
~
is t h e
identity
on
u
and
takes
(i,O)
to
2
,
(1,0)
to
O
. By definition v
UU
and
it
is e a s i l y
: ~U
verified
=
~{ = ~
that ¥ U
and
that
{~,uu}(y×2)
Formally, tions Any
can
such
is t h e
the
represented
~
is o f t h e
a
~ ~
: ~x~
and
bij k
=
Now
adjunctions
U
,,I U
: 2×2 ~ 2 x 2
.
adjunction functors
from
{pl~,p2~}
Since hence
natural
-x~ pi ~ =
2×2x2
where
has
transforma-
Pi
a right
{aio,ail}
to
~×~
.
: 2×2 ~ 2
adjoint,
it
where
. Similarly
: 2×2
~ 2
=
{bijo,bijl}
, so
{{{bloo,blo1},{b110,blll}},{{b200,b20I},{b210,b211)}}
functors
Boolean two
id
form
aij
where
are ^
~
as
i'th projection. pushouts
there
required
be
preserves 13
=
= 3
from
functions
projections
2×2 of Pi
two
to
~
are
the
variables.
: 2x2 ~ ~
where
same
There
as positive
are
i = 1,2
six
such;
, the
two
the constant
functors 'true'
: 2×2 ~ I
'false' and from
the
two
3 ~ 2
: 2x~ ~ ~
lattice by
°>2 ~1>~
operations,
, described
as p a i r s
of
functors
I,i
If
we
onto
then
'and'
=
'or'
= V =
A
=
abbreviate ~
by
ql
{{~o,2},{~o,~}}_
{{~,~I},{_2,~i}}
the and
q2
and
second
q2 =
{{'false''Pl}'{Pl''true'}}
four a d j u n c t i o n
projections
natural
transformations
=
{{{'false',pl},{A,V)},q2}
( 8 1 : ~ × ~ ~ u~)
=
{ql,{{pl,'true'},{A,V)})
(~2:~
=
{ql,{{A,V),{'false',pl)}}
=
{{{A,V),{p!,'truet}},q
(e2:!x ! ~ ~ )
~×~
means
the
All
of this
there
are
identity
structure
by exponentiation.
So,
on
transports
refering
.%
UC =
functor
itself
to the d i a g r a m
w
^
, Uc =
C ~
^
~
, LC = C
.) throughout at the
v
u
^
, LC = C
,,%
^ v = C3U c U C = U c U C = LcL C = LcL C --
adjunctions
w
LC
and a commutative
A
, LC- .......-4 ... L C
diagram
10
by
2}
~×~
satisfying
with
are g i v e n
functors
C ~
v
2×2x2
}}
~ 2x2)
~ ~x~)
of
; i.e.,
{{Pl''true'}'{'false''Pl
(~l:u~
1,1.7,
first
ql =
the
(Here
I0
u
Cat
end of
15
I,i
{£s,Su}
cl×A
This structure categories,
where,
(Fi,F 2)
~
c3Xa
"~
cAXR
gives rise to the properties
if
F i : ~i ~ ~
lim(_~I
Fi
=
BOo ,
We always treat
B{--
(FI,F 2)
via the two projections
.
' i = i,2
i~2
B 01
--
--
B
(
, .A 2)
as a c a t e g o r y over
~x~2
of the limit. A number of properties
of this construction will be found in [CCS] discussed
, then
F2 ~
of comma
in a different
§2. They are
form in Part III. We are inter-
ested in this w o r k in the following kinds of results.
I,i.9.
(Yoneda)
a i-I correspondence
If
F
and
G : ~ ~ ~
between natural
, then there is
transformations
~ : F ~ G
and functors
(B,F)
~ ~
~
/
(B,G)
(G,_S) rasp.,
%
_SxA Actually,
This
B_A- ~
(Cat,BxA_)
B__ A- ~
(Cat,A×B_~ op
is called a Yoneda
tion" of
/
two full and faithful embeddings
lemma because
is the bifibration
[CCS],
~ (F,B)
AxB
this determines
(~,F) -- B × ~
~
the c a t e g o r y
given by the "basic construc-
§5 applied to the functor
11
B(-,F(-))
: BZ p ~ Sets,
I,l
so the above full and faithful
12
functor
is equivalent
to the
functor B~A ~ S e t s ~ ° P x ~
taking
F : ~ ~ ~
to
~(-,F(-))
yields the o r d i n a r y Yoneda discussed
. Specializing
to
~ = !
,
lemma. This c o n s t r u c t i o n will be
for c a t e g o r y objects
in an arbitrary category
instead of Sets in II, §2 and for representable
%
2-categories
in Part IV.
I,l.lO. then there mations
(Adjointness)
If
F : ~ ~ ~
is a I-I c o r r e s p o n d e n c e
~ : ~ ~ UF
(F,B)
~
temp. ,
~
natural
The ways
and
/
satisfy the equations if and only if
for r e p r e s e n t a b l e
junctions between
This
~
2-categories
is discussed
I,i.il.
[CCS] and
(Fibrations)
inverse to
[FCC].)
IV.
fails for quasi-adfor expressing
in c a t e g o r y theory.
in I, §7.
if there exists a functor adjoint right
. This will
in Chapter
are important
forms of m a n y c o n s t r u c t i o n s
for a d j u n c t i o n
~ ~ = --I
2-categories
in which this c o r r e s p o n d e n c e
the global
(F,B)
A_x~
transformations
be discussed
transfor-
>
~
A_xB_ ~
,
e : FU ~ B ) and functors
(A,U)
/
Furthermore,
U : ~ ~ ~
between natural
CA,U)
)
~
(resp.,
and
P : ~ ~ B
is called a fibration
L : (~,P) ~ E J S = (P~2,E01)
A split normal
12
: ~
fibration
which ~
is right
(~,P)
(See
is a fibration
1,1
with a choice of
L
13
w h i c h satisfies certain equations.
II, §3). The c a t e g o r y of split normal age
fibrations and cleav-
(i.e., L) p r e s e r v i n g functors is isomorphic to
and the forgetful
functor
into
the fibration a s s o c i a t e d to Furthermore,
QF ( _ p ....~
A
_
(Cat,B~
F : A ~ B
the a d j u n c t i o n m o r p h i s m
(See
Cat BOP
has a left adjoint, being
(in
(B,F) ~ B
(Cat,B)
.
)
(B,F)
B_ has a l e f t - a d j o i n t left inverse p r o j e c t i o n of
(B,F)
I,i.12.
to
~
P --~ QF
.
(Adjoint Functor Theorem)
left a d j o i n t if and o n l y if
w h i c h is just the
PF
F : ~ ~ B
has a
has a left adjoint right
inverse. In
Cat
this leads to the usual adjoint functor
theorem by the following steps: i) Use the dual of Prop. 4.4 in [FCC] to show has such an adjoint
if and o n l y if each fibre
PF
(rB~,F)
has
an initial object. ii) An initial object
in a c a t e g o r y is an inverse
limit of the i d e n t i t y functor of the category. iii) Hence every
B ~ B
F
has a left adjoint if and o n l y if for
, lim((rB" •F) ~ ~)
exists and is p r e s e r v e d by
F . (J. Beck has called this the basic adjoint functor theorem.) iv) Add a solution set c o n d i t i o n saying that the categories assume
F
(rB~,F) preserves
have small
initial subcategories,
small limits.
13
and
I ,i
I,I.13. Given
~
(Kan e x t e n s i o n s )
14
Let
F
: ~ ~ ~
be
fixed.
the
Kan
, a functor
:
having
ZX
along
F
as
left
adjoint,
is c a l l e d
. It can be c o n s t r u c t e d
i) Let functors
K
[°Pcat,~]o
: ~ ~ ~
in the
left
following
be the c a t e g o r y
and w h o s e
morphisms
whose
extension
way. objects
are p a i r s
are
diagrams
G
A
>
A'
x
where
t
: K'G ~ K
is < G ' , t ' ~ < G , t > 2-comma
is left
rX~
: ! ~ ~ ii)
Here
=
it is the c o m p o s i t i o n
14
along
the
in
I,l
15
H. > [°Pcat,E]o - - 9
B
[°Pcat,X]
im x
where
H,
is given by c o m p o s i t i o n w i t h
iii)
In general, given
the a s s o c i a t e d fibration
A
i.e.,
, then
~(B',C')
FB,C I
[FB',C'
A(FB,FC)
A(Ff,Fg)
e
A(FB',FC')
commu te s. 2F2
(compatibility with units) r
IB
~
~(B B)
_~
'
i
FB,B "
ACFB ,FB) commutes. 2F3
(compatibility with composition) m(B,C) x~(C,D)
°
)
~(B,D)
iFB,D
FB,cXFc,D I A(FB,FC) ×A(FC ,FD)
o
)
A(FB,FD)
commutes. It is easily seen that the ordinary functor
.~(-,
-)
:A°PxA o o ~ Cat
lifts to a 2-functor,
21
denoted the same way,
Z,2
A(-,-)
and that
: AOPxA ~ Cat
in the alternative
Cat-natural
7
description,
transformation,
F(_ _)
as defined below.
becomes a
The partial
functors A(A,-)
: A ~ Cat
A(-,A)
: A°p ~ Cat
are called C a t - r e p r e s e n t a b l e A 2-functor functors (resp.,
FB, C
functors.
F : ~ -- A
is called
have properly
locally faithful.)
locally
P ~ e.g., F
if all
FB,c'S
P
if all
is locally full are full
(resp.,
faithful).
1,2.3. A Cat-natural [E-K]. Thus ~B : F(B)
a : F ~ G
transformation
is as defined
in
is a family of morphisms
such that the diagrams
~ G(B)
~(B,C)
FB'C
~
A(FB,FC)
GB,C 1
~A(I'~C)
A(GB,GC)
B
~(~B,i )
commute
for all
and
natural
transformation
,>
C . Alternatively, ~
o
: F
o
~ G
22
o
~(FB,GC)
it is an o r d i n a r y
such that the diagram
1,2
8
G °p ×G
o
o
,
oP× o
c(_,_)
Cat commutes;
i.e.,
(2.7)
[~(-,-) ( ~ P x l ) ]-G(_ _) = [A(-,-)
(A~)o
denotes
the c a t e g o r y of 2-functors
transformations
from
~
egory of a 2-category
to A
OB'
SB : ~B
MCN.
cat-
in which a 2-cell or m o d i f i c a t i o n
in
~(B,C)
and Cat-natural
X . It is the underlying
(the term is from Benabou) 2-cells
(1×~ o) ]'F(_,_)
s : ~ ~ ~' A
is a family of
such that the diagram
FB'C
*
A(FB,FC)
x(l,sc)
GB, C I A(GB ,GC)
&(FB,GC) A(SB,I)
commutes.
(Note that
Compositions nentiation gory,
A(l,s c)
are defined
is a natural
in the obvious
transformation).
fashion.
This expo-
yields a cartesian closed structure on the cate-
2-Cat ° , of small The usual
2-categories
full and faithful,
Yoneda embeddings
23
and 2-functors. locally full and faithful,
1,2
A ~ Cat AOp
are
easily
established.
a modification formations
~
of k
: A(-,f)
is a f a m i l y
transformations which
at
: Y ~ X
h
, one has
~
A(-,g)
X
the
is
if
~ =
locally
Cat-natural
by
c)
denote
to
A(X,B)
the
naturality
implies
that
component for
hk , then
of
small
~ = A(-,a)
, so Y o n e d a
by
2-categories,
2-functors
constitutes a 2-category
2-Cat.
If m o d i f i c a t i o n s
a 3-category
: Cat
i.e.,
adjoint
give
. Here
discrete
(see
~
rise (-)o
2-Cat
functors
are
1,2.6) , w h i c h
to a n a l o g o u s plays be
2-categories.
a bijection
between
the
the
LG
objects
: Cat
L~
on objects
and
~ 2-Cat
be
[(LG)~](A,A')
of
inclusion
Let
Cat
ones
role
and
which included,
we
denote
between I (-)I
Sets
in
2-Cat
and
. Let
of categories
: 2-Cat
o
and
~ Cat
as be
locally
"local
~ "~ o
and
[ ( L ~ o) A] (A,A') Let
~(X,A)
2-Cat.
I,I.5
LD
of natural
formula
transformations
obtains
The
Cat
A(X,g) }
then
trans-
denotes
: f ~ g
collection
we hereafter one
: A ~ B
full.
The
then
(~A) id A
f,g
(~X)h
h Thus,
~
from
. If
, then
if
of Cat-natural
: A(X,f)
functors
in
: X ~ A
A° p ~ C a t ~
instance,
{~X
between
is C a t - n a t u r a l
~X
For
9
= ~o[A(A,A') ]
"local
G"~
i.e.,
= G[~(A,A')]
24
a bijection
. In e a c h
case
on one
1,2
must c h e c k
that these
with compositions.
locally defined
colimits exactly
except
ucts of finite
the
products)
that
symmetric
since
izers,
as e q u i v a l e n c e of the
coequalizer
classes
of two 2-functors
of 2-cells.
1,2.4.
in
[ 7 ]) is w h a t
[CCS].
together
Thus
however,
of the
of two
that while description
in the c o d o m a i n 2-cells
of strings
in the and of
the p r o o f of 1,4.9).
a 2-natural
it is a family of m o r p h i s m s
25
Note
equal-
a v e r y careful
classes
qf
. (This
in the
it p r e s e r v e s
transformation
w i t h a family of 2-cells
Cat
coprod-
situation).
of m o r p h i s m s
involves
is c a l l e d
in
of the c o e q u a l i z e r
for instance,
A quasi-natural
between
has a c o n v e n i e n t
of e q u i v a l e n c e (See,
category
description
are
as in I,I.3
w h i c h holds
further,
of strings
a similar
lengthy discussion
(Bunge,
Note
of two functors
functors,
strings
category
categories,
and c o e q u a l i z e r s
interpreted
adjoint,
and
in the usual
the c o e q u a l i z e r s
closed
has a right
is as expected.
the c o e q u a l i z e r
unions
construction
so the u n d e r l y i n g
2-functors
are c o m p u t e d
is to be
monoidal
(-)o
of c l o s e d
same c o n s t r u c t i o n
(especially
is the point of Wolff's
and limits
in the sense
are d i s j o i n t
by e x a c t l y
everything
general
and c o c o m p l e t e
. Limits
coproducts
described
are c o m p a t i b l e
'! (-)0 ----4 LG
(2-Cat)-limits
as in I,I.3
fashion,
~ LD
is c o m p l e t e
are
operations
Then
L~ ° ~
2-Cat
iO
as
~
: F ~ G
.
transformation ~B
: FB ~ G B
illustrated
1,2
FB
Ff
......
GB
such
that
QNI
if
N : f ~ f'
QN2
II
Gf
, then
>
FC
"b
GC
af,-(GN)o B = ~c(FN)'af
= id ~I B
QN3
oB
~gf = Og ;]] ~f The
given
composition
of q u a s i - n a t u r a l
transformations
is
by
(2.9)
(a'G) B = a~o B Condition
af's
constitute
QNI
says
, (~'a) f = a ~ a f
that,
a natural
for
fixed
transformation
~(B,C)
FB'C
B as
and
C
, the
illustrated
~
A(FB,FC)
-5 '
A(FB ,GC)
(2.10)
A(GB ,GC) A(~B,I) Note,
however,
respect
that
to m o r p h i s m s
the of
t h e y do n o t c o n s t i £ h t e tions
in the c o n i c a l
of n a t u r a l i t y o(_)'s
are
that
the d i a g r a m
do n o t b e h a v e
B
C
and
. Thus,
a modification
diagram
is r e p l a c e d
the
a(_)'s
(2.7)
"compatible
with
26
taken
between
in 1,2.3.
by condition
naturally
the
together, two
Rather,
QN3 w h i c h
with
says
c o m p o s l 't l o' n " in the
composithe
role
that sense
1,2
~(B,C)×~(C,D)
12
{ gBC XtGc,D'tFBc x ...cCD ........ } ~
X]
%
jUX × jUX I ~(FB,GD) ~(FB,GD) ~
ju
(2.11)
;° ~(B,D)
ra
~
.
a(FB,aD) R
B,D commutes~ where
"jux"
denotes juxtaposition composition,
where natural transformations into functors into
(-)~
(-)
are regarded as
(notations as in I,I.4) and where
%
denotes the pullback of the two functors dlJux : A(FB,GC)2×A(GC,GD) 2 ~ A(FB,GD) doJUX : A(FB,FC)2×A(FC,GD) 2 ~ A(FB,GD) Similarly, condition QN2 says that the
~(_) 's
are "compatible
with units" in the sense that the diagram 1 (2.12)
~(B,B) ~
~
A(FB ,GB)2--
r~B ,B commutes. In terms of components, these assert the commutativity of the diagrams F(I B) FB ~
~C
~
FB
.
//
FB
(2.13) Gf
Gc
IFB
27
1,2
FUno(~,~ quasi-natural underlying 2-cell
denotes
such
that
(degenerate)
s
of
~ : f ~ f'
FB
2-functors
to
~ . It
Fun(~,X) between
2-cells
and
is the
in w h i c h
a
quasi-natural
{SB
is a 2 - c e l l
~ B'} in
: °B
in
~
, then
the
this
just
cube
Ff
FB
MQN.
of
~
: o ~ o'
is a f a m i l y
if
from
of a 2-category
or modification
transformations A
the category
transformations
category
13
/I
II
)
FC
Ff'
IIIII
I
cfc !
~rC
r
Gf
I
~ G C
sd
Jl GC
commutes. i-cells.
Gf'
Note The
of non-full
of
X
sub
these
be a s u b c a t e g o r y containing
all
for
are given
by
important
type
, (s'.s) A = s ~ ' s A
2-categories,
2-category
to r e q u i r e
of modifications
(s's) A = s~s A
Besides
~' o
it is s u f f i c i e n t
two compositions
(2.14)
Let
that
of
there
is an
Fun(~,X)
of
~
i-cells;
and
o
i.e.,
described A'
~' o
a sub
¢ ~
o
as
follows:
2-category
and
~
~ A
the
same
o
Then Fun(~,%; denotes with
the
i-cells
for a l l
sub
2-category
of
the quasi-natural
i-cells
f E ~'o , of
A,A') Fun(~,A)
with
transformations e A'
28
, and with
o
objects,
such
2-cells
all
that
1,2
modifications
14
of such quasi-natural
transformations.
We shall
use only two special cases i) Fun(~,~5'~%,~ o) o transformations formations.
is given by quasi-natural
whose restrictions
E.g.,
ii) Let
; this
to
~' o
are natural
F u n ( ~ , ~ o ; A , A O) = A ~
iso A
denote
the sub 2-category of
sisting of 2-cells which are isomorphisms. Fun(~,~';A, iso A) o such that if
consists
transformations
involving
~f
commutes
af .
1,2.5. The 2-comma c a t e g o r y of a pair of 2-functors
[FI,F 2]
F. : A. ~ ~ , i = 1,2, l l
2-category with objects pairs of the form f : FI(A i) ~ F2(A 2)
~% con-
Then
of quasi-natural
f ~ ~'o ' then the square
up to the isomorphism
diagrams
trans-
, morphisms
triples
is the
(AI, f,A2) (h i ,¥ ,h 2)
where in
of the form F I (h i)
Fi(A i)
....
F 2 (A 2)
and 2-cells pairs
F2 (h2)
(~I :hi~hl 'e2 :h2~h2)
f'F(@i) "¥ = ~-F(@2)f
(2.16) while
. Composition
(h~,y',h{) (hi,¥,h 2) the two c o m p o s i t i o n s
:
,~
~i(A[)
>
F 2 (A~)
of 2-cells
of morphisms
is given by
(h~hi, ¥' [~¥,h~h 2) ,
of 2-cells are
29
such that
1,2
(~1,~2)-(~1,~2)
(2.17)
15
= (~1-~1,~2*~2)
(~1,~2) (e1,~2) = (~1~1,~2~2) Alternatively, of
the
is t h e
[FI,F2]
FI
where
Fun
there
: [ F I , F 2] ~
can
be
PI~
= F
Ai
that
and
the
form
the
value
in 2 - C a t
with
where
~ Cat
by
example
QN
: ~ ~
F
F
of a 2-comma : A ~ Cat
. [I,F]
is t h e
. (See
[CSS].)
1,2,3 ~
of
: F ~ G
[F,G]
A
~ A
as p a i r s
(f,e)
: (A,a)
~
(B,b)
: F(f) a ~ b
Ln
F(B)
(g,~) (f,e) of morphisms
of
(f,idF(f)a) canonical
choice
category
and
I
such
that
is o n e
of
opfibred
and
the
category can
a ~ F(A)
where
be
and
takes over
described its m o r p h i s m s
f : A ~ B
. Composition =
: I ~ Cat
Its objects
where
is t h e
$
by
projections
conditions
a 2-functor
(A,a)
class
is i n d u c e d
transformation
as pairs
~
are
6i
P2 ~ = G
[I,F]
determined
F2 1 > ~ {------ A 2
where
and
a quasi-natural
important
!
. Hence
, i = 1,2
identified
An
~
' i = 0,I
Pi
say
6 Fun and
: ~ ~ ~
1,2.4
6 > ~ ~ o
_
~ = °PFun(~,°P~)
@i
The
limit
diagram
AI
and
inverse
in
is g i v e n
by
(gf,~-F(g)~) form
: (A,a)~
(B,F(f) a)
of cocartesian
morphisms
for
this
opfibration. Similarly, over
~
if
corresponding
F
: A9 p ~ Cat to
F
is t h e
30
then
the
category
fibration [I,(-)°PF] Op
,
1,2
16
where (_)op
: Cat ~ Cat
takes e a c h c a t e g o r y of this
fibration
morphisms
to its o p p o s i t e
are a g a i n p a i r s
f : A ~ B
position
in
: (A,a)~
A
and
The canonical
cartesian
1,2.6.
in the c a r t e s i a n
duals.
=
F(A).
Com-
are t h o s e o f the form
~
is a
(B,b)
(2-Cat)-category~
closed category
is n e e d e d
2-Cat
is a n o t a t i o n
T h e y are c o m b i n a t i o n s , where
A°P(A,B)
= ~(B,A)
b) o p ~
, where
°PA(A,B)
= ~(A,B) Op
op
A 3-functor
A , where is a
transformations, modifications,
t h e r e are v a r i o u s
A(A,B)
modifications
There
of such,
of small
and 3 - c e l l s
corresponding
to
the a n a l o g o u s
comma category.
Fun(A,~)
31
closed
3-categories.
k i n d s of q u a s i - n a t u r a l
constructions
for the seven
are 2 - C a t - n a t u r a l
rise to a c a r t e s i a n
3-Cat,
(see 1,2.3).
= °P[~(A,B) ]
(2-Cat)-functor.
giving
on the c a t e g o r y ,
op
i.e.,
of
~op
c)
in
(gf,F(f),~-~)
: (A,F(f)b)~
A 3-cateqory
In this w o r k all that
a)
(B,b)
e : a ~ F(f)b
morphisms
(f,idF(f) b)
possible
as above, w h i l e
is g i v e n by (g,~) (f,~)
enriched
(A,a)
The objects
are p a i r s (f,~)
where
category.
between
structure
Similarly,
transformations
and
. Here we n e e d o n l y
1,2
17
1,2.7. The 3-comma c ateqory 3-functors
F. : ~. ~ ~ , i = 1,2 l l
with objects and morphisms ing the notation 2-cells
[FI,F2] 3
is the three category
the same as in
of 1,2.5, a 2-cell
together with a 3-cell
of a pair of
~
[FI,F2]
is a pair in
~
F 2 (~2) f
(~I,~2)
F 2 (h~) f
y!
(2.18)
f'Fl (h I )
The compositions fashion.
Finally,
of 3-cells
>
f'Fl(e I)
of such 2-cells a 3-cell
are defined
is a pair
f'F1(h ~)
in the evident
(~I.~I-~1,~2:' ~ 2 ~ )
such that
(2.19)
y'(F2(~2)f)- ~ = ~'-(f'Pl(~l))¥
Alternatively, determines
a pair of objects
(Al,f,A i)
and
(A~,f',A~)
2-functors
12 (a2,A~) F2(-)f
> g(Fi(a I) ,F2(A2) (
and the 2-Cat-valued objects
horn object b e t w e e n
f'FI(-)
~i(al,a[)
these two
is the 2-comma category [F 2(-) f,f'F I(-) ]
It can also be described as an inverse
where
of
as illustrated:
>
F 2 (h 1) f
. Follow-
3-Fun
limit
is given by a "basic construction"
32
for
18
1,2
3-categories
as
in [CCS],
§5, but r e p l a c i n g
there b y a 2-cc~m~a c a t e g o r y
1,2.8. class of
A double
"morphisms"
g o r y structures; (~0,~1,-)
as above.
cateqory
carrying
i.e.,
the c o m m a c a t e g o r y
~
two d i f f e r e n t
a class
and
(~o,~i ,?)
i)
each
is a c a t e g o r y
ii)
~.~.
= ~.~.
(Ehresmann
M
with
[15])
is a
compatible
cate-
two structures
of d o m a i n , c o d o m a i n , a n d
composition
such that
1 j
]
and the objects
1
form a s u b c a t e g o r y ill) there
We shall o b s e r v e
in II,l.
is the same thing both
for the
first
structure. induced second
(Cf. also
"strong"
for d u a l i z a t i o n second
structure
one, w i t h
op~
(~,.~)
in
Cat
A 2-category
the p r o p e r t y
are also o b j e c t s
for the
category . However, is a special
that
for the
the c a t e g o r y
first
the o b j e c t s
structure
second structure
b y the
is discrete.
We g e n e r a l l y as the
-
object
[CSS]) ; i.e.,
on the objects
structure.
that a small double
c a t e g o r y with
structure
law
(~,.~)
are useful.
structure
structure
=
as a c a t e g o r y
interpretations
case of a double
for the other
is an i n t e r c h a n g e
(~,?~,).(~)
for one
think of the
or "horizontal" w i t h respect
structure
as the
dualization
33
structure
and w r i t e
to this structure~
is c o n s i d e r e d denoting
first c a t e g o r y
"weak"
while
©op the
or "vertical"
with respect
to it.
1,2
Double tions
functors
are described
properties
with
three when
category taken
gories
which
in t u r n m a y
category
one
functors
or m o r e
This
which
as
does
2-category
the double
category
same
shall
call
and dealt with
category.
and a morphism
do.
The
are
object
to a v o i d
o
or a l l
structures
is t h a t
34
of
some
of
to b e a b i than
a
"triple" to one
of triple
in w h i c h
2-categories.
two of
This
in 2-Cat,
is
so w e
confusion.
o
length
is a d i a g r a m
three
respect
only kind
= OPFun
an o b j e c t
but
cate-
refinement
rather
with
structures
triple
A further
i.e.,
category
2-categories,
(see II,i.)
at g r e a t e r
As d e s c r i b e d ,
two,
functors
structures
(Fun ~)
of
in o u r w o r k ,
occur
is~
category
section,
in t h i s w o r k
it t h i s
The category 1,2.5
in the n e x t
as a c a t e g o r y
always
- one,
discrete.
structures
that occurs
cases
m a y be
are only pseudo
the r e q u i r e d
category
an n - t u p l e
of the d o u b l e
not
categories
the
special
locally
transforma-
as h a v i n g
a triple
One defines
structures
be
way
form double
in t h i s w o r k
structure,
2-category.
of the
occur
category
is to a l l o w
which
are v a r i o u s
natural
structures.
this what
structures
There
the double which
from
to b o t h
t w o at a time.
similarly.
and double
in t h e o b v i o u s
respect
It is c l e a r
±9
(~,op~) in § I , 4
is a l - c e l l
mentioned
in
is a d o u b l e f : A ~ B
in
1,2
h
A
D
B
composition Another
being
one
structure
by defining
includes
we prefer
A(-,U(-))
the case
P
to
isomorphism
2-functors
2-functor
is
is e x a c t l y
2-category,
in the
U
(2.4)
fibrations,
the b r i e f
2-Cat-adjoint
: ~ ~ ~
only
a double
Fibrations
~(F(-) ,-)
Thus,
haw
3-Categorical
is a C a t - n a t u r a l
Cat-valued
and composition
object
and
diagrams.
set of d i a g r a m s
is a d o u b l e
obtains
[FCC]
F
of such
interchange this
,
as a c a t e g o r y
and
e.g.,
same
, miD = k
that
same w a y
We use C a t - a d j o i n t
on the
The
one
B'
composition
= h
composition.
is n e e d e d
between
A t
>
.............
k
horizontal
category
be v e r t i c a l what
)
I
:
given
20
that
. Here
(F!,F 2)
in 2 - C a t ~2
) ~(
(~,P)
exists and
~i
F2 A2
a
a S
is the c o m m a
of the d i a g r a m s
>Z(
35
for b r e v i t y ,
if there SL =
(4)).
is
21
1,2
and
S =
{I~2,~ ~I}
existence for Of
of
2-functors
f : A ~ B : JB
in
~
° f* ~ JA
clusion, E e gB with
: 22 ~
(8f) Eh'
= m
is e q u i v a l e n t
fibres,
, and C a t - n a t u r a l
m
then
and
the
JA
: D ~ E
there
P(h')
f*
mapping in
to the
: ~B ~
ffA
transformations
: ~A = p-l(A)
the u n i v e r s a l
l-cells
fh = P(m)
. This
between
, where
satisfying and
(~,p)
~
•
is the
property
•
and
is a u n i q u e
h'
h
that given
: P(D)
in
in-
~
~ A
with
= h
D
m 9
E
(Of) E
-
P (D)
~ B
(resp., then
2-cells,
there
(Note
that
[FCC].)
with
this
of
f*
a choice
2-category
with
ef
and
is c a l l e d
of such b e t w e e n
(Sf) E~'
with
= ~
satisfying
and
P
in
: ~ ~ ~
[CCS]~
cf.,
= ~B
' is
together
2-fibration.
The
and m o d i f i c a -
2-fibrations
over
~
is
~op isomorphic any PF
to the
2-functor : (~,F)
joint
-- ~
then
2-category it has
such
that
to the p r o j e c t i o n
possible
factorization
(2-Cat)
. If
an a s s o c i a t e d F = PFQF (~,F)
through
~ X
QF
this
a split-normal
36
: X ~ ~
is
2-fibration
where , and
F
,
= ~.)
(if p o s s i b l e )
2-functors
split-normal
P(~')
(idB)*
a split-normal
preserving
f~ = P(~)
and
stated
egf = SgSf
2-cleavage,
of c l e a v a g e
~ : h ~ h'
is i n c o r r e c t l y and
= f'g*
a split-normal
and
~'
property
, (gf)*
such
tions
is a u n i q u e
A choice
Sid B = id called
~ : m ~ m'
is r i g h t is the b e s t 2-fibration.
ad-
1,2
Analogous tions.
results
hold
The proofs later
are
note
for
then
the p r o j e c t i o n
in the easily
reference
3-category adapted
that
(~,F)
~
22
if A
~
case
from has
is a l s o
and
those
for o p f i b r a -
in
enriched
[FCC].
We
pullbacks,
a 2-fibration,
via
pullbacks. In fibration
I,I.13 gave
this w a s u s e d construction treat Let
rise
it w a s
observed
to a f u n c t o r
for K a n - e x t e n s i o n s . will
the d u a l P
ii)
be
case
: • ~ ~
needed since
: B ~
a split
normal
[°Pcat,E]o
In 1,7.14,
is w h a t
normal
will
, and
an a n a l o g o u s
for q u a s i - K a n - e x t e n s i o n s .
that
be a s p l i t
p
that
be u s e d
2-fibration.
Then
We there. there
is a 2 - f u n c t o r
: ~ ~
where that in
P(B) ~f
~
(i.e.,
and
goes
, then 2-cell
consisting
of
P(f)
the A,
[2-Cat,~] 3
are
other
defined
way).
: f, ~ f~
in 2-Cat)
and
(k,,id)
and
(Cf.
If
as
1,2,7)
in I,i.13
A : f ~ f'
is a C a t - n a t u r a l P(A)
is the
(except
is a 2-cell transformation
2-cell
in as
[2-Cat,~] 3
the m o d i f i c a t i o n
~f
illustrated
2-cell
P(~f) E = A .
JB
J B I f,
where
(~f)E
(See a l s o
J B tA,
is the u n i q u e
1,7.13).
87
such
that
1,3
1,3. Bicateqories.
I
The notion of a b i c a t e g o r y is c l o s e l y
related to that of a 2 - c a t e g o r y and m a n y of our main results involve notions description
from the theory of b i c a t e g o r i e s .
is e s s e n t i a l l y that of Benabou
1,3.1. A b i c a t e g o r v
~
The following
[BC].
consists of a set
(not neces-
sarily small), Ob ~ , of objects together w i t h BCI
(small) c a t e g o r i e s
~(A,B)
for each o r d e r e d pair of
objects, BC2
"composition"
functor s ~(A,B)×~(B,C)
? ~ ~(A,C)
for each ordered triple of objects , BC3
"identity" objects
I A ~ I~(A,A) I
BC4
"associativity" natural
o×I
> ~(A,C) ×~(C,D)
~(A,B) ×~(B,D)
>
"left and right identity" natural
!x~(A,B)
A ,
isomorphisms
~(A,B) ×~(B,C) ×~(C ,D)
BC5
for each
.........
>
~(A,B)
~
38
~(A,D)
isomorphisms
~(A,A) x~(A,B)
1,3
~(A,B)
2
I B .~ ~(A,B) x~(B,B)
1X ~
xl
f
~ ( a ,B)
,
subject to two conditions: BC6
The cube
(in w h i c h we w r i t e
AB
and
for
~(A,B)
>
ABxBDxDE
1
for all i d e n t i t y maps)
lxoxl
AB ×BC xCD ×DE
°/I
~A~C x 4
?xl
ACxCDxDE
o×I
ADxDE 1×o
lxo
1lxlx° 1XO
ABxBCxCE
ACxCE
AE
commutes.
(This is the
"pentagon" c o n d i t i o n
of the a s s o c i a t i v i t y isomorphisms.) BC7
The
(degenerate)
cube
39
for c o h e r e n c e
1,3
1 X
AA×AB×BB
lx
o
.......................
rI
AB×BB
3
×I× II
> AA×AB
-"eA~k o/¢
r n IB
.... o
III
~ ~ l
|~
IB .
IxAB
ABxl commutes.
1,3.2. between
A pseudo-functor
bicategories
together
(or m o r p h i s m )
is an o b j e c t
function
F F
: ~ ~ ~'
: Ob
~ ~ Ob
with
PFI
functors
PF2
natural
FA, B
: ~(A,B)
~
~' (FA,FB) ,
transformations o
fB(A,B) × ~ ( B , C )
c
>
I
o
~' (FA,FB) ×~' (FB,FC)
PF3
PF4
l-cells
~A
~(A,C)
: I F, A
two
conditions:
The
cube
~
~
F ( I A)
40
in
~' (FA,FC)
~' (FA,FA)
, subject
to
~'
1,3
4 I X°
ABxBC×CD x l /
ABxBD
FABXFBcXFcD FAB×FBD
AC×CD
o
> AD FAD
FAc×FcD i×o t (A)F(B) ×~'(B) F(C) x~ (C) F (D) ----c | e - - . . .
F (a) F(C) ×F(C) F ( D ) - -
o,
> ~ ( a ) F ( B ) xF(B)~'(D)
~ F(n) F(D)
commutes PF5
The (degenerate)
cubes l xAB
I
IIFAB
FAAXFAB
I
41
FAB
1,3
5
AB×!
ABxBB
AB
o
fABxI
FAB×FBB
FAB
Q
F(A) F(B) ×F(B)F(B)
-
o
A
commute.
A copseudo-functor except This are
the
can
~A'S
and
is c a l l e d
(F' ,~'
,~')
(F'F,qp"
can
~A"
tors
,~")
, then
strict)
be c o m p o s e d ,
a
If all
the
O's
pseudo-functor
pseudo-functor. where
given
: ~ ~ ~"
o F A ,B
= %0! FA,FB,FC =
F' (~A)
stacking
the d i a g r a m s
the c a t e g o r y Bicat
~eA,B,C
' ° ~FA
is a p s e u d o - f u n c t o r
by
duals.
direction.
b y the d a t a
,, ~ABC
We d e n o t e
of s t r u c t u r e ,
: ~B' -~ ~"
..{F'F~AB = F'F A , F B
this
suitable
(resp.,
Pseudo-functors
Vertically
by
sort
all go the o p p o s i t e
(resp. , i d e n t i t i e s )
a homomorphic
is d e f i n e d
same
~A,Bjc'S
a l s o be d e s c r i b e d
isomorphisms
then
is the
and of
in PF4
that
small
.
42
and PF5
composition
bicategories
shows
that
is a s s o c i a t i v e . and pseudo-func-
1,3
6
1,3.3. A quasi-natural transformation between pseudofunctors
F
and
F'
from
~
to
~'
consists of
QNP1
a family of l-cells
~B : FB ~ F'B
QNP2
a family of natural transformations FAB
~(A ,B)
> ~' (FA,FB)
q~J, ~t
(FtA,FtB)
.
aAB/]~'
~'(~A,1)
(I ,~B )
~, ~' (FA,F'B)
subject to two conditions: QNP3
the (degenerate) cube
FB i
FB
I (~B
aB
(BB) IB
/
/\
F'B
commutes for all
B .
43
I ~B
BB
I ~B ~ |
1,3 QNP4
The degenerate
7
cube
~g FA FC
~B ~A
~C
F~A
~,Igf~ commutes
for all
f
and
~ F , c g . Because of the non-strict
associativity
of composition,
as asserting
the commutativity
[ (F'g) (F'f) ]~A
this must be understood
(~'ABC)f,q~A
of the diagram >
F' (gf) ~A
I
~ T F A , F ' A , F t B ~ F t C
(F'g) [ (F'f)~ A] ~(F'g) ~f (F'g) [~B(Ff) ] ~gf
~ (~,FA,FB,FVB,F,C) -i [ (F'g) ~B ] (Ff)
~Og (Ff) [~C (Fg) ] (Ff)
I
aVFA,FB,FC,FtC acF (gf)
aC[ (Fg) (Ff) ] ~C (%°ABC) f ,g
44
1,3
8
The c o m p o s i t i o n of q u a s i - n a t u r a l
transformations
d e f i n e d by c o m p o s i n g such squares vertically, all the r e q u i r e d
instances of
MQNP.
inserting
~'
M o d i f i c a t i o n s are d e f i n e d as in 1,2.4~ is a family of 2-cells
again
is
{SA:~ A ~ ~
i.e., s : ~ ~ o'
such that
( ~ B ) f- [ (F'~)SA] = [SB(F ~) ]" (~AB) f
The two c o m p o s i t i o n s are d e f i n e d c o m p o n e n t - w i s e , bicategory
Pseud(~,~')
. Note that,
since the c o m p o s i t i o n s
o n l y use the c o m p o s i t i o n structure of 2 - c a t e g o r y then so is
Pseud
(~,~')
and yield a
~'
, if
~'
is a
. There are faithful
Yoneda e m b e d d i n g s ~ Pseud
(~°P,cat)
~op ~ Pseud
(~,Cat)
but they are not full in general. The c o r r e s p o n d i n g c o n s t r u c t i o n s yield a b i c a t e g o r y
coPseud
for c o p s e u d o - f u n c t o r s
(~,~')
1,3.4. Examples. I) Examples of b i c a t e g o r i e s
in [BC] that concern us are
a) m u l t i p l i c a t i v e c a t e g o r i e s
= b i c a t e g o r i e s w i t h one
object. The c a t e g o r y of a b e l i a n groups w i t h
®
as m u l t i p l i c a t i o n provides a simple example of a n o n - f u l l Yoneda embedding. b) Bim
= the b i c a t e g o r y of bimodules where objects
are rings w i t h unit, where 1-cells R-S-bimodules
RMs
phisms of bimodules.
and w h e r e
R ~ S
are
2-cells are h o m o m o r -
C o m p o s i t i o n is tensor product
45
1,3
over c)
the m i d d l e
Spans
X
from
V
2-cell
ring.
, where
An object
X
is a c a t e g o r y
of Spans to
9
V t
X
with
is a n o b j e c t
is a d i a g r a m
is a c o m m u t a t i v e
of
pullbacks. %
, a l-cell
V - X ~ V'
and a
diagram
X
I X t
Composition 2) T h e
construction
arbitrary have
The
of
i-cells,
multiplicative
~
in
by pullbacks. above
such
which
giving
composition
if o n e
Bim
bicategory
coequalizers
fixed
is g i v e n
are
c a n be g e n e r a l i z e d
that all categories
preserved
a bicategory
~
makes
category.
~(A,B)
by composition Bim(~
each category
An object
to a n
R
as
follows:
~(A,A)
~ ~(A,A)
with
a
is a m o n o i d
is g i v e n m o r p h i s m s mR
R
o R
which
are
involving
eR • R
associative "o"
a) A n o b j e c t A
, IA
~ Ob B
If
and unitary,
and
IA
of
Bim(~)
and
R
Similarly, ~(A,A)
~ R
and
is a p a i r
is a m o n o i d
composition ~(B,B)
on
~(A,B)×~(B,B)
~ ~(A,B) and
(A,R)
in
where
~(A,A)
determines ~(A,B)
~
£ ~(A,A)
isomorphisms
.
~(A,A) x~(A,B)
S
u p to the
actions
of
,
~(A,B)
R
46
~ ~(B,B)
are monoids
then
1,3
is a left
M ~ ~(A,B)
IO
R - right S-bimodule
if there
are given morphisms R o M
~
)M
, M ° S
~M
such that the diagrams
IBOM eR° M i
MoI A ~
M
MoS
M
)
RoM
MoSoS
~
M
MoS
m R oM RoRoM
RoM
~
Ro~
RoMoS
MoS
commute,
where we have
A morphism
left out
of bimodules
~
.....~...
~
M om S .....
> MoS
>
M
RoM
~
M
~,
e,
is a morphism
and
r .
f : M ~ M'
such that the diagrams RoM
)
R°f I RoM v
M
MoS
>
M
M' oS
~
M'
If >
MI
commute. b) A i-cell
from
b imodu I e
RMs
(A,S)
to
. A 2-cell
modules.
47
(B,R)
in
Bim (~)
is a morphism
is a
of such bi-
1,3
c) If
RMs
: (A,S) ~
ii
(B,R)
then the composition
and
of
M
sNT and
: (C,T) ~ N
(A,S)
is the tensor
product RMs ~ sNT where ~oN ~) MoN
M o S oN
.~M®N
Mo~
is a coequalizer 2-cells (B ,R)
S
in
~(A,C)
. The composition
is the induced morphism. is
That
of
The identity
for
RRR . R ~ M -~ M R
RoRoM
~
follows as usual since
mR°M Ro~
.~
•
eR~R~M .
RoM
.....
/
~eR°M
IAoR*M is a split coequalizer.
) M
7
/
IA~,M
Associativity
follows since
composition w i t h a fixed i-cell preserves
coequalizers.
Note that if the category of abelian groups, is treated as a m u l t i p l i c a t i v e
category via
i.e., as a b i c a t e g o r y with one object, Bim(Ab)
=
gory
V ~ X , then
Spans %(V,V)
whose objects
look like
are diagrams
48
then
Bim(Spans
%)
is the m u l t i p l i c a t i v e E
do dl
morphisms
~
Bim
3) In this work we shall be concerned with If
Ab,
~ V
cate-
~ and whose
1,3
12
>
E
where
both
as above
triangles
commute.
E'
The product
of two objects
is g i v e n b y
E×E t
2Y
V
"-%
E'
E'
V
V
and the unit as d i r e c t e d discussed tation
is
V
~ V
graphs with
. One can think vertices
in II,l and b i m o d u l e s
in terms
of
The c o n d i t i o n
Bim(Spans about
in
Bim(~)
associativity following If
X
then
no role.
easily verified
has u n i v e r s a l it satisfies
composition means
plays
that
in if
in II,2 and the %)
frequently However,
interpre-
of c o e q u a l i z e r s of the compo-
ignore
this w h e n
w e do note the
result:
coequalizers
the c o n d i t i o n s Bim(Spans
f : X ~ Y
%(V,V)
in II,2.2.
the a s s o c i a t i v i t y
. We shall
Spans
~ m o n o i d s w i l l be
the p r e s e r v a t i o n
in 2) above only affects sition
V
of
X)
(as w e l l
as pullbacks)
for a s s o c i a t i v i t y
. Here,
, then the
universal
functor
of
coequalizers
given by pull-
backs f* preserves topos
: (%,Y)
coequalizers.
since
there
f*
~
This has
(x,X)
is a l w a y s
a right
49
, satisfied
adjoint.
in a
1,3
1,3°5. illuminates (Cf.,
Fibrations.
various
other
use
aspects
of p s e u d o f u n c t o r s o f the
study of
and
spans
fibrations.
1,2.9.) i) H o m o m o r p h i c
in b i j e c t i v e chosen L
The
13
correspondence
cleavages~
: (B,P)
B
point
e Cat
from
(Cf.,
P
chosen [BC]
of view,
determines
B9 p
fibrations with
in 1,2.Q.
different
A category
with
equivalently,
~ _~22 , as
completely
pseudo-functors
to
Cat
: E ~ B
lifting
and
with
functors
[23]) . F r o m
consider
a "directed
are
a
~im(Spans(Cat)). graph"
BOo B01 and
the
Spans
structure Cat(B,B_~
(P,IdE) span
described . Consider
from
B
to
is a c o m m u t a t i v e
E
in I,I.6
makes
a functor
. An
P
"action"
this
a monoid
: E ~ ~
in
as a span
of this m o n o i d
on this
diagram
B2xE B
M
>
E
and
P
g where B2xE
the p u l l b a c k =
strict
(B,P)
. There
actions
cleavages modules L(f,E)
L
is f o r m e d
in the for
is g i v e n : M(f,E)
P
is a b i j e c t i v e sense . The
by setting ~ E
from
B 00
correspondence
of 3) a b o v e correspondence M = ~0°L
is d e f i n e d
morphism
50
! i.e.,
and
between
split
normal
from c l e a v a g e s
. Conversely,
as the v a l u e
of
M
to
given on the
M
,
1,3
....
f
id
in
L
B~
14
>
E
>
E
. Suitable c o m m u t a t i v e squares give the e x t e n s i o n of
to a functor
(B,p) ~ _~22 w i t h the d e s i r e d properties.
The
same c o r r e s p o n d e n c e gives a b i j e c t i o n b e t w e e n "2-actions", (i.e., actions w h i c h are a s s o c i a t i v e and u n i t a r y up to given c o h e r e n t isomorphisms)
and a r b i t r a r y cleavages.
are models of the 2 - t h e o r y of monoids~
Such actions
see 1,8. We thus obtain
a b i j e c t i o n b e t w e e n the following classes and indicated subc lasses : 2-modules over
fibrations over ~ chosen cleavages
(split-normal cleavages)
(modules)
homomorphic pseudofunctors (functors)
B9 p ~ Cat
The top line is d e s c r i b e d in greater g e n e r a l i t y on fibrations
with
in the sections
in the later chapters of this work.
ii) The c o r r e s p o n d e n c e s extend to isomorphisms of categories.
In each case, there are three n a t u r a l choices of mor-
phisms. T r e a t i n g
fibrations first, a c o m m u t a t i v e d i a g r a m
51
1,3
E_
T
_B
where and
P ~
and
P
are fibrations with chosen cleavages
gives rise to an adjoint
cartesian phism
to the identity
(resp., the identity). subcategories
chosen cleavages the possibilities (Fib~all)
(Cf.,
ST ~2 = TS
(~,~)
transformation
. (T,K) if
The results
A
is called
is an isomor-
of 1,6 show that
of the category of fibrations with
and all morphisms
(T,K)
as above. We denote
by )
u
(Split~all)
,~
preserving)
L
(cf., 1,6)
is the transpose natural
(resp., cleavage
these define
square
K•T=T
A : T~2L ~ LT
corresponding
_~
>
K
(~,P)
where
15
(Fib~cart)
)
(Fib~cleap)
u
~
u
(Split,cart)
[FCC], where different notation Turning now to modules,
~
(Split;cleap) is used).
defining
to a diagram
52
N = B~°k
gives rise
I ,3
2 B=xE
16
M
>
E_
B
This
N
is c o m p a t i b l e
a pair
(T,N)
or modules,
is called
associativity
A
and
~
is an
of i s o - g u a s i - h o m o m o r p h i s m s form c a t e g o r i e s
where
the
~,
We denote
(2-mod,quasi)
(Cf.,
k
functors
from
of 2-modules
From
or the
and h o m o m o r p h i s m s . N's compose
B__ °p
three cases
via v e r t i c a l
composition,
(2-mod,iso-)
(2-mod,homo)
)
U
(mod,iso-)
to the Cat
if
speak
by
)
to
one can
identity
All
U
turning
N
the p o s s i b i l i t i e s
~
(mod,quasi)
and such
is. In these cases we
U
Finally,
1,8).
isomorphism
if the c o r r e s p o n d i n g
of squares.
and u n i t s
a quasi-homomorphism
as the case m a y be.
reconstruct and o n l y
with
)
(mod,homo)
interpretation
, one c a n c o n s i d e r
in terms
of
diagrams
Cat where
K
is a functor,
functors natural all
or
functors
(i.e.,
~f's
are
all
and ~f's
F
and ~
can be h o m o m o r p h i c
can be q u a s i - n a t u r a l ,
are
identities).
F
pseudo-
iso-quasi-
isomorphisms) , or n a t u r a l
We denote
53
the p o s s i b i l i t i e s
(i.e., by
1,3
I C a t pseudo / ~ ~Ca~l quasi
17
) ~/ C a t pseudo / / rCa~ ) iso-q
U
a quasi-functor
: ~ ~ Fun(A,~)
~ Ob X
= Ff
of
to a square
>
required
: A×~
is a 2 - f u n c t o r b)
, ev(f,F)
Gf
satisfies Let
quasi-functor
= FA
Ff
immediately
of a
~ 2-Cat O
is t h e
gives
GA
it f o l l o w s
components
it is a 2 - f u n c t o r
FA
and
isomorphism
functors
(2-Cato)°Px(2-Cato)°Px(2-Cato)
Proof:
to a n
~ Fun(%,Fun(A,~)) ~
isomorphism
ii)
as
follows:
Ht(X)
= H(-,X)
to
, then
~
: Ht(x)
whose
[Ht(g) ]A = H ( A , g )
60
two
vari-
. By definition,
.
Ht(g)
transformation
of
~ H t ( X ')
components
are
1,4
and,
for a m o r p h i s m
f : A ~ A'
7
,
(HtX) A
(HtX) f -9 (HtX) A'
H(A,X)
H(f,X)
(HtX')A
. /> (HtX ' ) f
H(A,X')
H(f,X,) ~ H(A' ~X')
i.e.,
(HtX')A '
[Ht(g) If = yf,g
functor
from c)
%o
If
. It
to
Fun
o
~ : g ~ g'
is e a s i l y
seen
that
> H(A' ,X)
this gives
a
(A,~) is a 2 - c e l l
in
%
, then
[Ht(v) ]A = H(A,~) Clearly
Ht
is a 2 - f u n c t o r ,
e v ( A × H t)
= H
.
The then
uniqueness
follows
structive,
and amusing
set
the
as
% = !
in L a w v e r e
, ~
quasi-functors
, ~
, up to an
principles.
that
internal
is the u n i q u e
Fun(A,~)
from g e n e r a l
by deriving property,
of
and
[CCFM],
, and
22
of
p.9.
that
isomorphism,
can
also
Fun(A,~) That
from
X×~
be
from
inshown
this
is,
, and c o n s i d e r
of two v a r i a b l e s
such
It is i n t e r e s t i n g ,
the u n i q u e n e s s
structure
one
the to
structure ~
of
in these
cases. iii) level
of o b j e c t s .
formation t
Part
between
: H t ~ Hit
ii) If
gives ~
the d e s i r e d
: H ~ H'
quasi-functors
as f o l l o w s :
61
isomorphism
is a q u a s i - n a t u r a l of two
variables,
at the transdefine
1,4
(~t) X
:
Ht(x)
Jl
If
g
H
: H(-,X)
~
is a m o r p h i s m
in
: X ~ X'
, then
posed q u a s i - n a t u r a l Htx
H,t(x)
;i
(o(_)) x t i.e., ~X
~
8
~
t g
H' (-,X) Fun(a,~) is the m o d i f i c a t i o n between com-
t r a n s f o r m a t i o n s g i v e n by
Htg ~ H t X '
H(-,X)
H(-,g)
) H(-,X')
l
•
HftX t - - - ~ H ' ~ tx t It' g
l
H~ (-,X)
i.e., the c o m p o n e n t s are given by
Ht (_ ,g)'>" I-IV(- ,Xt )
(O~)A = ~A,g
" The cube
in 1,4.1, QN 2, can be interpreted p r e c i s e l y as asserting that t g
is a m o d i f i c a t i o n
a quasi-natural Finally, 1,4.i,
(I,2.4, MQN). Thus
let
s : a ~ ~'
be a m o d i f i c a t i o n as in
s t : ~ t ~ ~,t
iii). Then
Fun(A,~)
is the m o d i f i c a t i o n
sxt : Htx ~ H , t x
with c o m p o n e n t s
(s~)A = SA, x •
lished, once one has o b s e r v e d that
p r o d u c t s of
are functors 2-Cat
(as in
are the m o d i f i c a t i o n s
The n a t u r a l i t y of this c o r r e s p o n d e n c e
q-Fun(-×-,-)
is
transformation.
1,2.4) w h o s e c o m p o n e n t s in
~ t : H t ~ H 't
Fun(-,-)
is e a s i l y estaband
(but not 2-functors)
on suitable
and its dual. A proof of this for o
Fun(-,-) since
is suggested
in [CCS],
if, in the s i t u a t i o n
62
§6. A direct p r o o f is immediate,
1,4
9
F (4.2)
A'
......K.
>
A
~
~
H
> ~,
J
G is a q u a s i - n a t u r a l (H~K)A, = H(~K(A, ))
transformation, and
then so is
(H~K) f = H(~Kf)
a functor in the a n a l o g o u s
fashion;
HoK
, where
. q-Fun(-x-,-)
i.e.,
is
in the s i t u a t i o n
H
(4.3)
A' x~'
FxG > Ax~ ~
•
K
>
{,
H' where
F
, G
, and
K
are 2-functors, while
a
is a quasi-
natural t r a n s f o r m a t i o n b e t w e e n q u a s i - f u n c t o r s of two variables, it follows that and
Ka(FxG)
KH(FxG)
is a q u a s i - n a t u r a l
Alternatively, b e h a v i o r of F : At ~ A
is a q u a s i - f u n c t o r of two v a r i a b l e s
using this last fact, the functorial
Fun(A,~) and
transformation.
is d e t e r m i n e d since, g i v e n 2-functors
G : ~ ~ ~t
, then
Fun(F,G)
is the unique
2-functor such that the d i a g r a m
A' xFun (A, ~)
(4.4)
Fxl
AXFun (A, ~)
ev >
~%'xFun (F ,G)
ev
~%'×Fun (At ,~' )
>
commutes.
It follows d i r e c t l y from this d e s c r i p t i o n that
Fun(-,-)
is a functor.
~,
1,4.3. Remark. The n a t u r a l i t y of the i s o m o r p h i s m above u s u a l l y is used in the following three situations, the diagram on the left c o m m u t e s
83
in w h i c h
if and o n l y if the d i a g r a m
I ,4
I0
on the right does.
a)
T%xx
Yl
Fun ( i~, ~)
Gxx
~
iff
X
&x%H ' / ~
Ht~
Fun
un (G, ~5)
(4.5)
(&,~)
b) Fun ( A, ~5) Axx
iff H ~ ' ~
%
~
Ht~'-9 Fun(~.,~)
c) Axx
Z
&xG
~
1,4.4. There
is a natural
ii)
Fun(O,~)
iii)
Fun(A,lim Fun(lim
functor ~A
-~ I ,
embedding
Fun(l,~)
Xi,~)
C
z lim Fun(Ai,~) ev : X×~ A ~ ~
of two variables
considering
£5A ~ Fun(A,~)
~i ) -~ lim Fun(A,~i)
i) The 2-functor
Fun(A,~)
(4.7)
G
Corollary.
i)
Proof:
iff
t
which
is certainly
and hence determines
a quasi-
a 2-functor
is easily seen to be an embedding
2-functors
from
22
64
into
~A .
by
1,4
ii) Since for all
~
il
is not o n l y initial, but empty, ~ × ~
is e m p t y
% , so there is a unique q u a s i - f u n c t o r of two
variables
~×% ~ ~
for any
~ . Thus
I
Fun(O,~)
versal m a p p i n g p r o p e r t y c h a r a c t e r i z i n g a r g u m e n t shows that
Fun(~,~)
satisfies the uni. A similar
2 ~ .
iii) Since limits in 2-Cat can be r e p r e s e n t e d as suitable subobjects of products,
it is evident that a q u a s i - f u n c t o r
of two v a r i a b l e s H
:
A×%
lim
~
~.
~-
1
c o r r e s p o n d s to a u n i q u e l y d e t e r m i n e d cone of q u a s i - f u n c t o r s of two v a r i a b l e s H.
l
:
AxX
-~ ~,
1
This shows that the unique q u a s i - f u n c t o r of two variables, e--v , m a k i n g the d i a g r a m
Axlim Fun(A,~i)
ev .........)
lim ~i
pr. 1
Axpri I
AxFun (A, ~i )
ev.
>
~i
1
commute
satisfies the same u n i v e r s a l p r o p e r t y as
ev : AxFun(A, lim ~i ) ~ l~m ~5i , and h e n c e Fun(A, lim ~i ) -~ lim F u n ( A , ~ i) C o l i m i t s are a bit more difficult. right adjoint,
it p r e s e r v e s colimits.
Since
-×%
has a
The proof consists
in
showing that there are natural c o r r e s p o n d e n c e s b e t w e e n the following types of m o r p h i s m s and cones:
65
1,4
a)
% ~ Fun(l~m
12
Ai,~)
b) (lim ~i )xx ~
c) lim (Ai×x) ~ d)
cones
{AixX
~
~}
e)
cones
{X ~ F u n ( A i , ~ ) }
f)
X ~
lim Fun(Ai,~)
In b) , c) , a n d
d),
From
1,4.3
one
1,4.2
and
and
the
of two
properties
variables
are
intended.
of ordinary
limits,
This
has a)
~-* b)
so w e m u s t trivially it
quasi-functors
show for
, ~-~ that
~-~ f)
since
to consider
to b y p a s s
,
b) , c) , a n d
coproducts
is s u f f i c i e n t
easiest
e)
c)
and
are
equivalent.
they
are
disjoint
coequalizers.
show
F
d)
b)
~-* d ) .
unions,
In this
case,
holds so it is
Suppose
p
G is a c o e q u a l i z e r be
in 2 - C a t
a quasi-functor
We must
show
there
of
(see
two
1,2.11,
variables
is a u n i q u e
K
I).)
such : ~×X
Let
that ~
~
H
: ~×X ~
H(F×%) with
KP
= H(G×%) = H
.
Define
Now as
if
K(Q,-)
= H(B,-)
: X -- •
K(-,X)
: n ~ C
is t h e
q
a string
: Q ~ Q'
and
(fl ..... fn )
H ( f l , X ) , .... H ( f n , X )
g
where
PB = Q
functor
induced
: X ~ X'
, then
of morphisms
is c o m p o s a b l e
66
in
in •
q ~
. We
by can such set
H(-,X) be
represented
that
1,4
Yq,g
Using Cat,
Yfn,g ~] Yfn-I ,g
=
the e x p l i c i t one
desired
shows
1,4.5.
description
that this
unique
Theorem:
of c o e q u a l i z e r s
There
and g i v e s
is a s t r i c t l y
quasi-functor
of two v a r i a b l e s
This
is the m a i n
fact we w a n t
about
Fun(X,~)
(see 1,7).
. There
as follows:
is fixed,
then
on the right
is
Similarly,
But g i v e n
~
and
T
on
this c o m p o s i t i o n . can be
F
H
G
K
seen that
Ha
: HF ~ HG
and that c o m p o s i t i o n
with
a 2-functor : Fun(~,~)
composition -oF
as functors
the s i t u a t i o n
transformation
Ho-
since
surrounding
for the c o m p o s i t i o n
it is e a s i l y
gives
means
to d e s c r i b e
formula
Consider
is a q u a s i - n a t u r a l H
"Natural"
are three ways
i) An e x p l i c i t
H
which
o > Fun(A,~)
quasi-adjunctions
If
strictly
in all t h r e e v a r i a b l e s
for all of the c o m p l e x i t i e s
given
the
associative,
it is r e s p o n s a ~ l e
2-Cat °
in 2-Cat and
of two variables.
Fun(~,~)×Fun(~,~) Proof:
" " " [~ Yfl 'g
is w e l l - d e f i n e d
quasi-functor
unitary composition natural
13
with
F
: Fun(~,~)
~ Fun(A,~) on the left gives
a 2-functor
~ Fun(A,~)
as indicated,
67
instead
of a c o m m u t a t i v e
1,4
square one has,
for a n y
A
14
£ A , a diagram
H (~A ) >
HF (A)
HG(A)
(3.8)
/ KF(a)
>
KG(A)
K(aA) One checks
easily
that
Yg, T
is a m o d i f i c a t i o n
T
making
: Ka- TF
and satisfies
ii) C o m p o s i t i o n variables
~(-)
TG" H~
the r e q u i r e d
is the unique
properties.
quasi-functor
of two
the d i a g r a m
AxFun(~,~) ×Fun(~,~)
evxl
) ~xFun(~,~)
(3.9)
~xFun (~, ¢) commute.
From
this,
unitaryness
as w e l l
uniqueness.
The unit
corresponding this,
to the
the strict
>
isomorphism
functor
to make
68
from
! ~ Fun(~,~)
A×~ ~ A . However,
quasi-functors
in a n y case).
and strict
are e a s i l y d e r i v e d
is the u n i q u e
one m u s t d i s c u s s
¢
associativity
as n a t u r a l i t y
(Actually this m u s t be done tivity
ev
to do
of three variables.
any sense out of a s s o c i a -
1,4
iii)
The t h i r d
approach
15
is to s h o w that t h e r e
is a
tensor product with Fun(~,~) and t h e n p r o c e e d
z Fun(A,Fun(~,~))
as in [E-K],
b o t h of t h e s e p r o c e d u r e s ,
1,4.6. n ~ 2 , H :
Definition.
n ~ A. ~ • i=1 i
5.10.
omitting
We shall b r i e f l y detailed
proofs.
i) A q u a s i - f u n c t o r
consists
describe
of n - v a r i a b l e s ,
of q u a s i - f u n c t o r s
of two
variables H ( A 1 ..... A i _ l , - , A i + 1 ..... A j _ I , - , A j + I ..... A n ) : X i x A j ~ for all which that
i < j
and all c h o i c e s
a g r e e on o b j e c t s
of i n d i c a t e d
and as 2 - f u n c t o r s
for all t r i p l e s of i n d i c e s
fi the d i a g r a m
i < j < k
extraneous
variables (fi ,1,1)
H (A i ,Aj ,Ak)
,
H(ai,A~, ~ )
.......
Ak E ~
of l - v a r i a b l e ,
: AI• ~ A !1 ' f 3 : A 3 ~ A 3t ' f k (in w h i c h
objects
such
and morphisms : Ak
~
~
are omitted)
;
~(A[,Aj,~)
> ~(ai,A~,a k)
(1,1,f k)
Aj ,~)
H ( A i ,Aj , ~ )
H ( A i , 3, , A
)
69
,
1,4
16
commutes. ii) A q u a s i - n a t u r a l family
of quasi-natural
transformation
~
: H ~ H'
is a
transformations
{aA 1 .... , A i _ 1 , - , A i + I , .... A n } for all
i
a n d all c h o i c e s
quasi-natural choices
of
fications denoted
transformations
indices
iii)
of objects
A modification
s : ~ ~ ~'
the preceeding
that
instance,
morphisms a tesserac
are
there
for a l l
is a f a m i l y o f m o d i -
The resulting
2-category
is
A i,C)
to b e
are no
taken
justified,
"higher
are e q u a l .
on quasi-natural
cube)
no
categories,
whose
if f o u r
then one obtains
3-dimensional
ways.
further
These
and can
relations".
of quasi-functors,
in t w o d i f f e r e n t Similarly,
it m u s t
commutativity
from d i f f e r e n t
(4-dimensional
1,4.7.
n H i=l
in the d e s c r i p t i o n
can be composed
faces
two composed
conditions
are
required
transformations.
Theorem.
i)
If
functor
of
n.-variables 1
and
functor
of
n-variables,
then
ni F. : H A. ~ ~. is a q u a s i l j =1 lj l n G : E ~. ~ • is a q u a s i i=l 1
n.
G ( F i ..... Fn)
: .~i( ~ i= 3=
Aij ) -- C
n is a
are
i < j
for e a c h v a r i a b l e .
For
shown
cubes
, which
by
Remark.
For
~ Ak
o f two v a r i a b l e s
qn-Fun(
be
Ak
q u a s i - f u n c t o r of
~ ni-variables. i=l
70
1,4
ii) If n+i-variables, Ht
n i=l
n H : A× H A. ~ • i=i l
17
is a q u a s i - f u n c t o r of
then there is a unique q u a s i - f u n c t o r
Ai -* F u n ( . ~ , l g )
of
n-variables
such
that
the
diagram
n Ax H A. i=i I
AxFun (A, ~)
>
ev
commutes.
iii) This c o r r e s p o n d e n c e extends to a natural isomorphism n q n + l - F u n ( A x i =~i Proof.
n A.I,~) Z q n _ F U n ( i =~i
~i'Fun(A'~))
The proof is a s t r a i g h t f o r w a r d b o o k k e e p i n g exercise
b a s e d on the p r o o f of 1,4.2. This now justifies the second a p p r o a c h to the p r o o f of T h e o r e m
1,4.5.
We n o w turn to the third a p p r o a c h using tensor products. This is based on the o b s e r v a t i o n of S. Mac Lane that quasifunctors should be special p s e u d o - f u n c t o r s , information
together with the
from J. D u s k i n that Benabou has given a u n i v e r s a l
c o n s t r u c t i o n to "straighten out" p s e u d o - f u n c t o r s .
The idea
of the c o n s t r u c t i o n was then w o r k e d out in c o n v e r s a t i o n w i t h D u s k i n and Mac Lane. A l t h o u g h w i t h Mac Lane's
it is not needed
later, we b e g i n
s u g g e s t i o n since it can be u s e d together w i t h
1,4.21 to give an a l t e r n a t i v e c o n s t r u c t i o n p r o d u c t w h i c h follows.
71
for the tensor
1,4
1,4.8.
Proposition.
18
There
is an inclusion
(see 1,3.3)
q-Fun(Ax~,~) C-~coPseud(Ax~,~)
Proof.
Let
Define a
H : ~x~ ~ ~
be a quasi-functor
copseudo-functor
H : Ax~ ~ •
of two variables.
by the following data:
a) A(A,A') x~(B,B')
~ ( A '
,B')
I
H(A,-)×H(-,B')
C(H(A,B) ,H(A',B'))
~(H(A,B) ,H(A,B')) x~(H(A,B') ,H(A',B'))
i.e., H(f,g)
b)
= H(f,B')H(A,g)
e(f',B) (f,B) e(A,g') (A,g) @(f,B') (A,g)
= id = id =
id
~(A',g) (f,B) = Yf,g
i.e.,
the only n o n - p r e s e r v a t i o n
of composition
is in the
situation H(f,B)
H(A ,B)
~(A,g)
>
H(A' ,B)
¥
~(A,B')
H(f,B')
IH(A' ,g)
H(A' ,B')
We have chosen to represent
this as a copseudo-functor
than a pseudo-functor
yf,g
with
72
rather
in the lower triangle because
1,4
19
it then looks like a commutation Yf,g which helps
to keep things
calculations to
(H,~)
H(f,g)
:
~ H(A',g)H(f,B)
straight.
that the conditions
transformations
1,4.0. Theorem. 2-categories,
a ® ~
y
are exactly equivalent
The c o r r e s p o n d e n c e
and m o d i f i c a t i o n s
There exists
of [E-K],
q-Fun(X×~,~)
A ~ I A]
and
isomorphism
II,5.iO.
(See 1,4.23 also). We construct
The objects of
for
~ Fun(~Fun(~,~))
the conditions
and apply 1,4.2,
for
is then clear.
a tensor product
together with a natural
Fun(~,~) satisfying
One shows by explicit
on
being a pseudo-functor.
quasi-natural
Proof.
relation
A ® ~
so that
~ Fun(~,~)
iii). A ~ ~
B £ I~I
of "approximations
are pairs
(A,B)
. The morphisms
of the diagonal"~ (fl,gl) (f2,g2)
...
of objects where
are equivalence i.e.,
classes
strings
(fn,gn)
where i) f i e
exist;
i.e., ii)
morphism.
A , gi ~ ~
and the c o m p o s i t i o n s
flf2
"'" fn
glg2
"'" gn
O f i = 01fi+ i , 0og i = 01gi+ I .
for all
i , either
fi
Two strings are equivalent
the smallest equivalence
or
gi
if they are made so by
relation compatible
73
is an identity
with c o m p o s i t i o n
1,4
20
such that (fl,l) (f2,!)
~ (flf2,1)
(1,g I) (1,g 2) ~ (l,glg2) Composition Note that
of m o r p h i s m s 1
is induced by juxtaposition
always denotes whatever
identity map
of strings.
is appro-
priate. The 2-cells of First of all, there
A ® ~
are c o n s t r u c t e d
for all n o n - i d e n t i t y
l-cells
as follows.
f ~ A
and
g ~ ~ ,
is a 2-cell : (f,i) (1,g)
Yf,g Now consider
equivalence
~
(l,g) (f,l)
classes of w e l l - f o r m e d
strings
A = [k I .... ,An] where
Ai
is either
k i = (Ti,~ i) in
~
¥f,g
where
and either
~i T.
for some
f
is a 2-cell
in
or
~,
1
iii) A string
and
g , or
A , ~i
is a strong
is a 2-cell
identity.
1
is w e l l - f o r m e d
if w h e n e v e r
(Ti,~i) (Ti+l,~i+l)
then
TiTi+ 1
(Ti,~i) ¥f,g
then
Tif
yf,g(Ti+l,~i+l)
then
fTi+ I
then
ff'
and
and
~i~i+l
aig
kiAi+ 1 =
¥f,gYft,gl are defined
in
A
and
~
iv) Two w e l l - f o r m e d
and
g~i+1 gg'
respectively. strings are equivalent
made so by the smallest equivalence juxtaposition
and
of strings such that
74
if they are
relation compatible
with
21
1,4
Strong composition
(T,1) (T',I)
~ (XT',I)
(1,~)
~
(I,~')
(1,~')
is induced by juxtaposition of strings.
The strong domain and codomain are given by @IA = @Ikl
~oA = @okn
and
where $i(Ti,~i)
= (~iTi,~i~i)
~iYf,g = (~if,~i g) The weak domain and codomain are given by ~i A = [~iAl ..... ~iAn ] where
~i(Tj,~j)
= (~i~j,~i~j)
OoYf,g = (f,i) (i,g)
and , ~i¥f,g = (i,g) (f,!)
Square brackets denote the equivalence class of the indicated l-cell. Finally,
the 2-cells of
strings of well-formed
A ® ~
are equivalence
classes of
strings
= Cai.~.....A n] such that
~O~i = ~IAi+l
. Two such strings are equivalent
they are made so by the smallest equivalence
relation
with composltlon ' ' " such that v)
¥f, g(f,l)-(f',l) ¥f,g
¥flf,g
(l,g') ¥f,g'Yf,g' (l,g) ~ yf ,g'g
vi)
(l,g') (f',l)yf,g-yf, ,g,(f,l) (l,g) ¥f,,g, (l,g) (f,l) - (f' , I) (l,g') ¥f,g
when
f'f
and
g'g
are defined.
75
if
"compatible
1,4
vii)
If
T : f ~ f'
, a : g ~ g'
yf, ,g,.(T,1) (1,~)
viii)
when
T'-T
The w e a k
(~'-T,1)
(t,~')-(1,~)
~
(1,~'-o)
domain
position" and
~
, and of
then
~'-~
are
defined
and codomain the w e a k
strings~
in
A
are g i v e n
composition
i.e.,
it is r e p r e s e n t e d
then
~ (1,a) ( T , 1 ) - ¥ f , g
(T',I)'(T,1)
and
= ~lAi
22
F-F'
and
by
$
, respectively.
~ F = ~ A O o n
is i n d u c e d
is d e f i n e d
if
by
,
"dot
juxta-
~1 F' = ~ F
by
* l " " 2 " " " " "*n" * l " " " " " * ~
To d e f i n e 0.A. i 3 F
the
strong
is i n d e p e n d e n t
and
F'
domain
and codomain,
of
so we c a n
j
are r e p r e s e n t e d
F = [AI'...'A and as
~1 F' = follows:
~oF
, then
if
n ~ m
weak
string
of
identities
~i F = ~iA1
, F' = [ A I ' . . . ' A
strong
. If
m]
composition
FF'
is ~ e p r e s e n t e d
, let
AI ~ be a n y
set
that
by
]n
the
one o b s e r v e s
length into
m
....~m
constructed
F . This
still
by
inserting
represents
F
suitable b y viii).
Then
~' If
n > m
Finally,
make we can
"compatible with
both
with
=
Ec ~ p - . . . ' ( ~ ) J
the a n a l o g o u s explain
that
composition"
compositions
construction
for
the e q u i v a l e n c e means
described
76
that
here.
F'
relation
being
it is to be c o m p a t i b l e
1,4
We
leave
a 2-category interchange law for
matter
squares
of
how
FF'
of the d e t a i l s
to the r e a d e r . law
requirement
most
vi)
(I,2.1,
above.
There
(2.2))
described
identities
, which
23
of c h e c k i n g
is a n o n - t r i v i a l
corresponding
in 1,2.1,
Using are
this
(2.4). , one
inserted
is n e e d e d
to
that
case
to the This
shows
in g i v i n g
this
is
of the
interchange
is c o v e r e d
that
by
it d o e s n ' t
the d e f i n i t i o n
show associativity
of s t r o n g
com-
position. There
is an o b v i o u s J
given
by
the u n i q u e
(~,IB)
so l a b e l e d
2-cell.
2-functor
tivity
of the c u b e
transformations
The tivity our
: ~ ® ~-9C
with
H = HJ
Q N 2 in 1,4.1,
ii)
~ : H ~ Ht ~
shows
correspond : H ~ H'
H
, with , there
. The
that
7f,g is a
commuta-
quasi-natural
bijectively
. Similarly,
to q u a s i -
modifications
other.
conditions
of the d i a g r a m s
order
(IA,T)
for a n y
H
to e a c h
, J(A,-) (T) = Clearly
transformations
correspond
of two v a r i a b l e s
: A x ~-->A ~
J(-,B) (~) =
unique
natural
quasi-functor
in
[E-K],
II,5.10
(in t h e i r
require
terminology
the
but
commuta-
adapted
to
of v a r i a b l e s )
((A~B) ~C ,D)
P
(A@(B~C) ,D)
>
>
(B~C, (A,D))
P
> P
77
(C, (A~B ,D) )
(C, (B, (A,D)))
1,4
(I~A, B)
24
P
~
(A, (I,B))
(A,B)
where
a
and
~
are induced by the c o r r e s p o n d i n g d i a g r a m s
at the level of u n d e r l y i n g categories. Fun(X,Y)
. The second d i a g r a m
d e r i v e d explicit e x p r e s s i o n equivalent
([E-K],
v i t y isomorphism
II,4.1)
Here
(X,Y)
means
is immediate from the e a s i l y
for
e
. The first diagram
is
to the c o h e r e n c e of the a s s o c i a t i -
a . This,
in turn,
is e q u i v a l e n t
in this
case to the fact that the various ways of w r i t i n g a tensor product of four v a r i a b l e s all r e p r e s e n t q u a s i - f u n c t o r s of four variables~
and this,
finally,
is e q u i v a l e n t to the non-
existence of "higher c o m m u t a t i v i t y relations"
(Remark after
1,4.6) , whose proof we have omitted, m o s t l y because the relevant d i a g r a m
is too big. A l t e r n a t i v e l y ,
the first d i a g r a m
above can be shown to commute as a diagram of 2-categories by reducing everything
to q u a s i - f u n c t o r s of several variables~
e.g., one easily e s t a b l i s h e s "p"
isomorphisms c o m p a t i b l e w i t h
: Fun(A~D~)
,~) ~ q-Fun(Ax(~(9~) ,~) C q3-Fu n (Ax~x~,~) Z q-Fun((~)
x~,©)
Z Fun((~49~) ®~,~) This finishes the c o n s t r u c t i o n of the tensor product.
78
1,4
The metry to
of
final
question
and this
is a n e n r i c h e d
"hom"(A,~)
= Fun(~,~)
its r e l a t i o n
in fact, This
been
to
is t h e
a hom-functor
(-)
strongly
adjoint
~ Fun(A,-) with
respect
. The discussion Fun(X,~)
for t h e
lack of sym-
is
adjunction
incorrectly
accounts
treated
that what we have A e
and
to be
A ® ~ . One wants
- ~ ~ . Note
25
of this other
is r a t h e r
described
seemingly
to
intricate,
adjoint and has,
by me
on several
occasions.
excessive
generality
in w h a t
follows. The basic choice
of
on t h e
left.
consideration
squares with
the
is c o n n e c t e d
2-cell
directed
with
our consistent
upwards
(u)
as
>
A 2-cell
directed
right. ness, (e
This we
downwards
is c l o s e l y
can also
for e q u a l ) .
follows
we
We
always
first
We
set
FUnd(A,~ from
~
to b e to
~
forming
the c a s e treat
the d i a g r a m
duals.
of commuting
the case
, a = u
Definition.
FUne(~,~)
to
like
of
on the
For completesquares
Fun(-,-)
In w h a t
set = d
1,4.10.
related
include
looks
(d)
= ~A the
, and e = e
Let
and
X
~
FUnu(A,~)
2-category
, whose
and
whose
morphisms
79
be
2-categories.
= Fun(~,~)
objects
are
are q u a s i d - n a t u r a l
. We define 2-functors
1,4
transformations; i-cells
i.e.,
{~A:FA
~
~ GA}
26
: F ~ G
together
consists with
a family
a family
Ff
FA
of
~>
FB
>
GB
of
of
2-cells
Qd N
GA
satisfying 2-cells 1,2.4,
Gf
conditions
analogous
are modifications
P!oposition.
°PFunx(°PA,°P~) x = u
: ~ ~ ~'
, d
There
The
an obvious
, e
case
: A ~
for
the
~
x = e
We
treat
and T° p
: A'
~ A
and
in
A° p
and
2-cell
and whose
to those
in
the case
in
°PA $
~op
f
families
category
the wrong
way.
of Hence
80
and
the
has
2-cells
we
write
that
same
: oPF
~ op~
f : A ~ A'
corresponding
the
op~
If
: oPA
if
clear
is
various
oPF
a 2-cell,
is t h e n
: F ~ G
there
in d e t a i l .
Similarly,
for
. It
cases,
in t h e
x = u
transformation
- being
x ( A, ~) o p
objects
: AoP ~
: f' ~
modifications - go
isomorphisms
= Fun~(~,~)
In a l l
T : f ~ f'
transformation
as a quasid-natural
natural
20PFun
2-functors.
A
fop
the FoP
corresponding in
are
is t r i v i a l .
between
, we write
is a i - c e l l
natural
1,2,3,
. In p a r t i c u l a r ,
bijection
2-categories. F
QN
analogous
= Funx(A°P,~°P)°P
F U n x ( o p A o p ,op~op)
Proof.
1,2.1,
MQN.
1,4.11.
where
s
to
a quasi-
description
~ oPG
in t h e
l-cell
, but
that
codomain
°PFunu(°PA,°P~)
F U n d ( A, ~)
1,4
For the case o f strong a °p
: F °p ~ G °p
: G Op ~ F °P l-cells
in in
{(gA )Op
{gA : FA ~ GA} fop
: B ~ A
G°PB
F u n ( A ° P , ~ Op)
in
~
A Op
a morphism
is the
. This
~ F°P(A)}
- together
consists
in
with
same as a m o r p h i s m
~op
G°PA
(~B)opl
- i.e.,
2-cells
Ff
FA
I(~A )
~
fop
i-cells for
/ > FoP (fop)
~A
F°PA
GA
are u n a f f e c t e d
) FB
~
op
Modifications
of a family of
as i l l u s t r a t e d
GoP ( f o p ) )
F°PB
dualization,
Fun(A°P,~°P) Op
: G°P(A)
in
27
Gf
~B
"~ GB
by this d u a l i z a t i o n ,
and h e n c e
FUnu ( xop, ~op) op ~ FUnd ( A, ~)
We next
s t u d y the v a r i o u s
possibilities
tors of two v a r i a b l e s
and q u a s i - n a t u r a l
them.
logical
There
are e i g h t
for q u a s i - f u n c -
transformations
possibilities
depending
between on
!
whether
the
"down".
In fact o n l y six occur.
case
yf,g'S,
the
~A s, and the
are
include
"up" or
the c o m m u t i n g
for c o m p l e t e n e s s .
1,4.12,
Definition.
i) A g u a S i u - f U n c t o r is a q u a s i - f u n c t o r variables yf,g
We a g a i n
~B1S
and these
of two v a r i a b l e s .
H : ~x~ ~ C
is "down"~
o_~f tW ° v a r i a b l e s
i.e.,
y's s a t i s f y
is d e f i n e d yf,g
A quasid-functor as in 1,4.1
: H(A',g)H(f,B)
equations
analogous
81
H : A×~ ~ of two
i) e x c e p t
that
~ H(f,B')H(A,g) to QF 2 of 1,4.i
, i).
1,4
A quasie-functor
of two variables
ii) A quasi functors
y,z
-natural
of two variables
a) quasi
Y
-natural
H
28
H : A×~ ~ •
transformation and
H
is a 2-functor.
between
consists
quasi x
of
transformations
(r(_) ,B : H ( - , B ) b) quasi -natural
--~I(-,B)
transformations
z
~A,(-) for all
A
in 1,4.1
ii) , QN 2. Note
natural"
means
vious
and
B
: H(A,-)
satisfying
~ H(A,-)
conditions
that "quasi u
"natural".
analogous
means
Modifications
to those
"quasi"
and "quasi e-
are defined
in the oh-
fashion. iii) q-Fun
The cases
u~d,u
x~y,z and
(Ax~,~) d~u,d
denotes
the resulting
are excluded.
(I.e.,
2-category.
there are
25 rather than 27 cases). 1,4.13.
Proposition.
There are natural
i) q-Funx~x,z(~×~,~)
-~ FUnz(~,FUnx(A,~))
ii) q-Funx~y,x(AX~,~)
-~ Funy(A,FUnx(~,C))
iii) q-Funx ~y 'z(AX~,~)
-~ q-Fun-x~z ,y(~XA,C)
isomorphisms
op [q_Funx ~Y, z (Op A×op ~, opc) [q-Fun- - -(A°P×~°P,~ Op) ]op x~y,z iv) F u n x ( ~ S , ~ ) Proof.
The proofs
adaptations numerous follows
-~ q-Funu~x,x(aX~,C)
of i) , ii) and iii)
of the proofs
diagrams
of 1,4.2
are straightforward
iii)
like the one in 1,4.1
from the observation
and 1,4.11,
ii). The proof of iv)
that objects
82
using
on the left are
1,4
2-functors from
X ~ ~
to
•
29
w h i c h c o r r e s p o n d to quasi u-
functors of two v a r i a b l e s from
Xx~
to
• , while m o r p h i s m s
on the left are quasi -natural t r a n s f o r m a t i o n s b e t w e e n such x 2-functors w h i c h are e a s i l y seen to c o r r e s p o n d to quasix, xnatural t r a n s f o r m a t i o n s b e t w e e n the c o r r e s p o n d i n g q u a s i u - f U n c tors, using the explicit structure of
1,4.!4. Theorem.
A ~ ~ .
The tensor p r o d u c t of 2 - c a t e g o r i e s is
part of a m o n o i d a l closed c a t e g o r y structure on 2-Cat variable
(a b i c l o s e d c a t e g o r y in the sense of L a m b e k
o
in each
[28]),
i.e., & ®
(-) ---4 FUnu(A,-)
; (-) ~ ~---4FUnd(~,-)
and there are natural isomorphisms i) F u n u ( ~ , ~ )
Z F u n u ( ~ , F U n u ( A , C ))
ii) F U n d ( ~ , ~ ) Furthermore,
~ FUnd(A,FUnd(~,~))
the e n r i c h e d r e p r e s e n t a b l e
structures commute;
i.e.,
iii) FUnu(~,FUnd(~,~)) Proof. with
~ FUnd(~,FUnu(X,~))
i) is the same as 1,4.9. x = d
functors for these two
ii)
follows from 1,4.13 iv)
together w i t h the special case
ii). The a d j u n c t i o n s
follow b y c o n s i d e r i n g these
at the level of objects. Finally, i) and ii) , taking
u;d,d
(x;x,z)
iii)
= (u;u,d)
=
of 1,4.13
isomorphisms
is immediate
from 1,4.13
(x;y,x). Note that i) , ii)
and iii) also follow from a d j o i a t n e s s plus the a s s o c i a t i v i t y of the tensor product
(end of proof of 1,4.9) .
1,4.15. Definition.
Let
A
be a 2-category.
Fun X = FUnd(~,~)
83
Then
I ,4
1,4.16.
30
Corollary.
i) F U n u ( A , F u n
~) 2 FUnd(~,FUnu(A,~) Fun (Fun u (A, ~) )
ii) Fun u(A~2,~)
~ Fun u(~,Fun u(A,~)) 2 °PFun(°PFunu(A,~))
iii) FUnd(2®A,~)
Z Fun(Fund(A,~))
iv) Fun(Fun ~) 2 FUnd(2_~2,~ ) Not____~e. In particular, natural from
transformations)
A
to
Fun ~
can be given with
morphisms
2--2
with 2-functors
Fun(A,~)
(i.e., quasi-
can be identified with 2-functors
or from
for the 2-cells
in the
in
X ® ~ in
to
~ . A similar analysis
Fun(A,~)
by replacing
formulas above. Thus 2-cells can be identified from
A
to
FUnd(~2,~)
or from
X ® ~2
to
. If one wishes, be eliminated
by using
1,4.11,
that
of
is a special case of 1,4.14 ii) , the definition
op~ = ~
since
~
special cases of 1,4.14
1,4.17.
can
in i) is a special case of 1,4.14
and the second is the definition
from 1,4.14
d
from 1,4.14 and 1,4.16.
Proof. The first equation
in ii)
all subscripts
of
Fun(-)
iii)
. The first equation
i) and the second follows Fun(-)
and the property
is locally discrete,
iii) and iv) are
ii).
Corollary.
There are isomorphisms
o P A ~ op~ _~ o p ( ~ ~ A) Aop ® ~op _~ (~ ® A)°P opAo p ~ op~op _~ op( A ® s)op
84
of 2-categories:
1,4
Proof.
It follows
are natural
from 1,4.11,
31
1,4.14,
and 1,4.13 that there
isomorphisms
OPFun u ( O p ( ~ )
,°P~)
~ FUnd(~,~)
q-Funu;d,d(A×~,~) op [q_Funu ;u ,u (°P~×°PA' opt) ] z °PFu n ( o p ~ o p ~ u Setting
op~ = o p ( ~ )
opt)
or
°P~°PA
op(~)
produces
2-functors
: op~op~
such that the effect of the above isomorphisms
on objects
is
given by composition
with these functors.
verse to each other.
(I.e., the isomorphism part of the Yoneda
argument works for natural transformations able functors).
The second
and the third follows
1,4.18.
Remark.
isomorphism
is proved
similarly,
The various versions of the D-Yoneda [E-K],
it is called the representation
[9], Kelly
between D-represent-
from the first two.
lemma are available here; e.g., where
Hence they are in-
[24], and Day-Kelly
1,8.6~
I,I0.8;
theorem.
II,7.4,
Parts of Dubuc
[8] are also available;
those parts which only use the biclosed
structure
namely
and not
symmetry.
1,4.!9. A
and
from
~) , Fun 2-Cat
to
Proposition. x
(A,-) 2-Cat
(resp., q-Fun
x;y,z
A
(~x~,-)
, and the isomorphisms
14, and 15 are 3-natural. Proof.
For all 2-categories
(resp.,
is a 3-functor
in 1,4.9,
il, 13,
(Cf. 1,2.6.)
It is easily checked,
using the explicit
part i) of the proof of 1,4.5,
formula
that the transpose
85
in
of composition
1,4
yields
a factorization
..........
>
F u n (~, ~)
which
gives
cases
are h a n d l e d
beyond tors
the
2-category position
3-functor
and
for our
said
Suppose
them
however,
1,3).
respect
K
the
Pro-
accordingly.
a similar
and q u a s i - n a t u r a l
1,4.5
there,
trans-
of the
discloses A, ~, a n d
are p s e u d o - f u n c t o r s Then
already
a fatal •
and
are ~
there
is a p r o b l e m
: Pseud(A,~ since,
when
~ Pseud(A,~) H
86
is a p p l i e d
and
in t r y i n g
to d e f i n e Ho(-)
ana-
it is e s s e n t i a l
an e x a m i n a t i o n
transformations.
func-
standard
. Thus
out
In fact,
for
in the n o t a t i o n
of t h e s e
to the
to c a r r y
other
that nothing
in 2-Cat °
However,
formula
. The
the b e h a v i o r
pseudo-functors (see
Fun(A,-)
m u s t be c o r r e c t e d
to be able
F, G, H, a n d
quasi-natural
with
[CCS]
to do so.
that
about
structure
o f the e x p l i c i t
bicategories, are
be nice
on
Note,
variables
281 o f
between
purposes
analogue
T
can be
for b i c a t e g o r i e s ,
formations
structure
3-category
on p a g e
Fun (A, ~) Fun (A'~)
Fun (Fun (A, ~) ,Fun (A,~))
analogously.
first
It w o u l d
flaw.
>
naturality
in t h e i r
lysis
32
to a d i a g r a m
1,4
Ff
FA
gets
FA'
GAv
GA
one
33
Gf
a diagram
HFf
HFA
HFA'
Ff,~ A H~ A v
Ha A
HGA
In g e n e r a l , get
HG f
there
is no w a y
a quasi-natural
no good
to t r y
to g e n e r a l i z e to
or
path
other,
a whole since
there
to p u t
transformation
transformations
even
J~
allow
of
the
is no w a y
from notion
diagrams
2-cells
together HF
these to
HGA v
2-cells HG
to
. It d o e s
of quasi-natural
like
from
one
to c o m p o s e
87
composition such
diagrams.
to
the
1,4
There T : H ~ K
is no a n a l o g o u s
, since
one
has
34
difficulty
in d e f i n i n g
HFf
>
HFA '
T~ ..... )
KFf
no d i f f i c u l t y ,
1,4.20. said
for all
Definition.
to be d e f i n e d
that
for all
if
H
KFA'
In the
situation
above,
Ha
is
if
(Ha) f = Note
TFA,
f .
~ F f , O A ' : (HOA,) (HFf) is i n v e r t a b l e
where
the d i a g r a m s
HFA
with
TF
f . We
~ H(OA,(Ff))
set
(~Ff,OA,) -i (H~f) (~aA,G f)
is h o m o m o r p h i c
then
Ha
is d e f i n e d
for all
u .
1,4.21. i) between
If
H~
situation
is a q u a s i - n a t u r a l
is d e f i n e d
transformation
iii) homomorphic iv) on a n y
: HF ~ KF
In the
above,
transformation
pseudo-functors.
ii) natural
TF
Proposition.
(-)F
then
between
: Pseud(~,~)
H~
: HF ~ HG
is a q u a s i -
pseudo-functors.
~ Pseud(~%,~)
is a s t r i c t l y
~ Pseud(~,~)
is a p s e u d o - f u n c t o r
pseudo-functor. H(-)
: Pseud(A,~)
sub-bicategory
of
Pseud(A,~)
88
on w h i c h
it is d e f i n e d .
1,4
35
1,4
36
( t , i ) (o,~)
(t,1) (O,t)
>
(t,i) (o,t')
(~,1) ( o , t ' ) /
/
(t',ll(o,t)
(t',i)(o,~)
> (t',~) o , t ' ) Yt,t t
Yt,t Ytt,t ~ ¥tw,t
(1,t)(t,O)
(I, N) (t,o)
.> (1,t') (t,o)
O)~ (1,t) ( t ' ,o)
(i,~)(t',o)
Only relation vii)
)
of 1,4.9.
most of the diagonal
(i,t')(t',o)
is needed here. We have omitted
compositions
in this picture of
2×2×2
.
From these two cases one can more or less visualize ~ ~
and
~ ~ ~2
for arbitrary
seem out of reach at the moment. Cat ® Cat
~ . More complicated cases For instance,
~ i.e., describe Cat ® Cat((~,~) , (~' ,~') )
90
what
is
1,4
at least
in special
Another associative regarded Then
cases.
interesting
case
~ ® ~
by strings
~
has a single
object
XI
Ai ~ ~
, Bi ~ ~
is e m p t y or
or
. The
B. ~ B!
l
l
Ai
and
w i t h a single
B
,
object.
and its m o r p h i s m s
are r e p r e s e n t e d
"'" An Y n
and
2-cells
Y
is e m p t y or
n
are g e n e r a t e d
together
with
B
n
, and
by morphisms
interchange
2-cells
AB ~ BA
l
to the r e l a t i o n s
calculate
~
~
of s t r i c t l y
of objects
where
subject
c at e g o r i e s ,
and
XIBIA2B2
l
is that of a pair
and u n i t a r y m o n o i d a l
as 2 - c a t e g o r i e s
A. ~ A!
37
specific
in 1,4.9.
examples
T h e r e has
of the m o n o i d a l
not b e e n time to category
that arises
from this c o n s t r u c t i o n . 2) The c o n s t r u c t i o n to
(-)~ . In I,i.5, ~x(-)
to the Kleisli
category
terms of
~ ~
between (-)
1,4.15,
iii). ~ ®
for this
the two
functors
above,
them.
It is easier
: ~ ~ ~ ~ 2 : t ! ~ (t,i) (o~t)
6i : Z~
~ ®~
: t,
>(1,t)(t,o)
91
has
a much
and a C a t - n a t u r a l this
properly,
structures
represented,
O
is isomorphic
to d i s c u s s
transfering
by 6
Fun(-)
analogous
is a triple,
Cat t
two triples
has two c o t r i p l e
~ ~ ~ ® ~
(-)~
and
triple.
namely,
, the s t r u c t u r e (-)
that
is a cotriple,
complicated structure~
transformation
is, of course,
we r e m a r k e d
or e q u i v a l e n t l y
more
Fun(-)
in
by
given b y
in the t e r m i n o l o g y
I ,4
Ytt
represents
This
induces
a quasi-natural
a Cat-natural
38
transformation
transformation
between
between
6
these. ~
(-)
O
and
~i ~
functors 2-Cat
(-)
or e q u i v a l e n t l y
from
Fun(Fun(-))
is a 2 - c a t e g o r y ~
Fun(Fun
~
- i.e.,
~ Fun
~ " Ytt
1,4.23 of P
: A ~ A
then has
there
such
F
that
if
is a u n i q u e simple
functions
n
finite
In]
F
: ~ ~ ~
which
=
such
is n a t u r a l
The
problem
(I,3.2)
with
is p r e s u m a b l y
Let
of Benabou
I : ~ ~ Sets
(Note:
and
,
a special
construction
{1,2 ..... n }
maps.
FP = F
take
increasing
[0] = @ ). A s e t
a functor
sequences
: n ~-~ S e t s ( I ( n ) , X )
of elements
a unique
Seq X
over
of
and
sequence
X
~
whose
such that
objects ~
~
(x i ..... x n)
(x i ..... x n)
(cf.
92
I,i.ii,
and
are
: m ~ n
map
(X~(i) ..... Xa(m)) for e a c h
which
two
is a n y c o p s e u d o - f u n c t o r ,
follows.
a fibred category
determines
~
between
1,5.5).
(unavailable)
set
2-functors
is a C a t - n a t u r a l
be a 2-category.
solution
AO p ~ S e t s and hence
(cf.
: A ~ ~
to the c o r r e s p o n d i n g
determines
are
, a 2-functor
for e a c h
A
precisely:
and a copseudo-functor
It is as to t h e
~ Fun(-)
~ ~ a 2-cell
2-functor
o f the c o r r e s p o n d i n g
the o r d i n a l
in
Let A
for b i c a t e g o r i e s .
X
A.
a 2-category
a relatively
case
~
such a t h i n g
ADDendix
finding
for e a c h
. This means
Fun(Fun(-))
Fun(Fun(-))
transformation
yields
the t w o c o r r e s p o n d i n g
Fun(-) and
~ , natural
is a C a t - n a t u r a l in
to
Fun(-)
(by 1,4.19) | a m o r p h i s m transformation
between
1,2.9.)
1,4
If Seq(Ob
A
and
A)A,B
be
B
are o b j e c t s
the
full
by sequences
(X i ..... X n)
Note
A ~ B
that
if
or
I
n = 0 if
A = B
then
If
a
2-category and
such
and
corresponding
there
~(A,B)
for
a sequence
n > i
and,
to d e s c r i b e
~)
and
determined Xn = B
sequences for
n = 2
sequence
(A)
for
then
are
the
fibred
~
same
category
.
with
(A,B)
, while n = i
is the
small
as t h o s e
in
over
A
Seq(Ob
X)
A,B
~ ) A , B ]°p ~ C a t
(X i ,X 2 .... ,X n) if
A , let
functor
[Seq(Ob taking
no such
2-category,
is the
Seq(Ob
Xi = A
sequence
the o b j e c t s
to the
are
2-category
of
that
is a u n i q u e
is a small
that
such
a unique
there
in the
subcategory
then
in w h i c h
39
A = B
the v a l u e s
of
to
, taking this
n-I [[ 2~(Xi ,Xi+ 1) i=± (A) to I . It is s u f f i c i e n t
functor
on m o r p h i s m s
of the
form
A
i) in w h i c h
(X i ..... X i ..... X n) case
~
it is the p r o d u c t
(X I ..... X i ..... X n) of i d e n t i t y
functors
with
c o m p o s it ion A(Xi_ i,X i) x A ( X i , X i + i)
° > A(Xi_ 1 , x i + i)
;
and ii) in w h i c h
(X i ..... X i , X i ..... X n) ~ case
and
(i+l)'st
and
then
it is g i v e n factors
taking
by
in the
the p r o d u c t
(X i ..... X i ..... X n)
inserting product
~
for
of
identity
: -I ~
A(Xi,Xi)
between
the
i'th
( X l , . . . , X i, .... X n) functors
with
the
functor
Ixi In p a r t i c u l a r , satisfied
(A,A)
~
by composition
(A)
gives
and u n i t s
93
IA IX
: ! ~
X(A,A)
, show
that
. The this
relations extends
1,4
40
to a functor. In the c o r r e s p o n d i n g "cartesian"
fibred category,
A(A,B)
there
are
morphisms
(fl ..... fi+Ifi ..... fn ) ~
(fl ..... f i ' f i + i .... 'fn )
(fl ..... f i ' I x i ' f i + i ..... fn ) ~ for e a c h c o m p o s a b l e
sequence
Given a triple
(fl ..... f i ' f i + i ..... fn )
of l - c e l l s
of o b j e c t s
from
A,B,C
in
A
to
B .
A , there
is a
functor Seq(Ob
A)B, C ~ Seq(Ob A)A, c
~A,BxSeq(Ob
( (Xi ..... Xn) ' (Yi ..... Ym ) ) ~ which
induces
a unique
"cartesian"
(Xl ..... X n = Y i ..... Ym ) functor between
fibred
categories A(A,B) xA(B,C) which
is the
cartesian
morphisms.
an o b v i o u s of
A(A,B)
in 2
(~A,B,C) f,g
over
IA
on e a c h
This
~(A,B)
in
the s e q u e n c e
such cartesian
and
the d e s i r e d
A . There
is
the u n i q u e : B ~ C
is the g i v e n c a r t e s i a n of
, so
eA : P(IA)
~
~ ~A
sequence
,
morphism.
as the
The
fibre
is the g i v e n
morphism.
morphisms
~A'S
f : A ~ B , g
inclusion
(A)
in
the g i v e n
g i v e n b y the i n c l u s i o n
is the
Since composition cartesian
for
(f,g)
A(A,A)
P : a ~ A
as the fibre o v e r
in w h i c h , : gf ~
fibre and p r e s e r v e s
is the c o m p o s i t i o n
copseudo-functor
of l e n g t h
unit
inclusion
° ) A(A~C)
above
is a c a r t e s i a n
can be e x p r e s s e d
by composition. universal
It f o l l o w s
property.
94
in terms
functor, of the
from this t h a t
all of the eA,B,c'S P
has
1,4
NOW let in
A . Let
~(A,B)
Z
41
be any family of composable
ZA, B
be the set of all cartesian morphisms
in
of the form (fl ..... fi+ifi ..... fn ) ~
where
(fi,fi+l)
~ Z
and let
(f! ..... fi'fi+i ..... fn )
are made
(i.e., c o e q u a l i z e d with their domains).
respect to the c o m p o s i t i o n ~[[~-i]]
which
F : A ~ ~
is stable with
defined above and yields a 2-category
F(f,f')
In particular, Z
This
identity
is universal w i t h respect to copseudo-functors
such that
the class
be the c a t e g o r y
~(A,B) [[~A,B-I]]
in which all of these cartesian morphisms maps
pairs of i-cells
A ~ ~
= F(f)F(f')
is given by taking
of all pairs of the forms
((i,g) ,(i,g'))
and
for all in
(f,f') X ×
((f,i) ,(f',l))
((f,l) ,(!,g))
. Then
® ~ = ~x~[[~
]
,
A simple example of this construction
is
checked by the observation
that copseudo-functors
from
!
to
Cat
X
and
transformations
B. Then
Fun(A,~)
if
~f
~ : F ~ G
iso
FUnx(X,~)
is
structure
given by considering
(or, later,
simply
quasi-natural
for all
We
i ) for this
is an iso-quaSix-natural
is an isomorphism
Iso-Funx(X,~)
is another biclosed
in which all 2-cells are isomorphisms.
shall use the prefix case. Thus
! = AZ p • This
are the same as cotriples.
I,~.2~ Appendi~ between
of Benabou
~ Z .
transformation
f
is the locally full sub 2-category of
determined
= Fun(X, Ao;~,iso
~)
and iso-quasi-natural
by these. Note that (I,2.4).
Iso-Funu(A,~)
Iso-quasi-functors
transformations
95
between
=
of n-variables
them are defined
1,4
analogously.
A
® ~ iso
42
is c o n s t r u c t e d by inverting all the
¥f,g
2-cells If
~ : F ~ G
is an iso-quasi -natural transformation, U
~
then
~A = ~A
i s an i s o - q u a s i d - n a t u r a l t r a n s f o r m a t i o n , where -1
and
~f = ~f
This yields an isomorphism
iso-Fun u(A,~)
which
-~ iso-Fun d(A,~)
is c o m p a t i b l e with the c o m m u t a t i v i t y
lSO
in w h i c h
yf~g
isomorphi~n
lsO
is sent to
(¥g,f)
-I
, and shows that this
c o n s t r u c t i o n gives a symmetric monoidal closed c a t e g o r y structure on 2-Cat
o
1,4.25 A p p e n d i x C. Let
2-Cat®
Cateqor.ies enriched ~n 2 - C a t @
denote the m o n o i d a l
(non-symmetric)
closed cate-
g o r y w h o s e objects and m o r p h i s m s are 2-categories and 2-functors,
respectively~
is g i v e n by
Fun(-,-)
C a t e g o r i e s enriched w a y as in
[E-K]
equivalently,
and whose
with
in
internal h o m - f u n c t o r
its associated
2-Cat
are defined
@
. Note that since
yf,g
in the usual
~A ¢ Fun(A,~)
since there is a 2-functor
taking the 2-cells
tensor product.
, or,
A ® ~ ~ Ax~
of 1,4.9 to identities,
an o r d i n a r y
3 - c a t e g o r y m a y always be considered as a 2-Cat®-category. In 1,7 we shall need the notion of a q u a s i - e n r i c h e d between
2-Cat -categories. @
Let
A
enriched h o m - f u n c t o r s denoted by Then a q u a s i - e n r i c h e d
functor
function,
F
denoted by
and A(-,-)
F : A ~ ~
~
be such, w i t h and
z(-,-)
is an object
, together w i t h 2-functors
g6
funct0r
1,4
FA, B
A(A,B)
:
~ ~(FA,FB)
for all ordered pairs of objects @A : IFA ~ F(IA) A,B,C
in
43
A,B
in
, and for all ordered
& , quasi-natural
A , l-cells
triples of objects
transformations
eA,B,C
as
illustrated
A(A,B)
c,
® A(B,C)
FA, B (9 F B ~ C I
~(FA,FB)
satisfying x
q~A,B, ~ l
is replaced by
of PF4 and PF5 of 1,3.2
Q . Note
between 2-categories
stances,
to 2-Cat~-bicategories.
and since
2-cells
in
it will be used
g "composition"
it means.
h
........-...~ B
-~------~
k is a diagram
g7
functor
is rather
in complicated circum-
A
f
AI ~
their
is the
There is an obvious extension
we illustrate p r e c i s e l y what
composable
functor
as 2-Cat(D-categories
Since the notion of ~uasi-enriched complicated
in which
that a ~ a s i - e n r i c h e d
regarded
same as a pseudo-functor.
FA'C
> ~(FA,FC)
® Z(FB,FC)
the analogous
of these notions
) A(A,C)
C,
Given two
44
1,4
in
A(A,C)
hf
h~
>
hg
kf
k~
~
kg
. F
the subscript
and A,B,C
e
then consist of maps on
~fh : F(h)F(f) and 2-cells properties,
(omitting
~ )
~ F(hf)
~h~ ' eTf ' etc., satisfying
some obvious
plus the property that the diagram
F(h) F (f) ~h/~///
F (h) F (~)
>
F (h) F (g)
IF(T)F(f) F(T)F(g)
F (h~!
F(hf)
F(Tg)
F(Tf) F(k) F(f)
...F(k) . F(g)
F(k) F(g)
F(¥e
~ F (kf)
/"~kf ,~- F (kg)
F (k
i
= 1,2,3
~t
1
then the diagram
[FI'F2]
i% 2
I
(5.8)
[F1,F 3 ]
[F2'F3]
(VI'W'V2)
V%
j
(V2'W'V3)
(Vl ,w'v3)
commutes. ii) Let transformations.
~ : G1 ~ F1
and
~ : F2 ~ G2
be quasi-natural
Then
[~,~] : [FI,F 2]
>
[GI,G 2]
(5.9) {P1
is the 2-functor determined formation
AlXA 2
by the composed quasi-natural
in the diagram
107
trans-
1,5
8
P2 [FI,F 2 ]
>
(5. lo)
%2
~ - ~ I G2 PI eFIF F1
I G1 i.e.,
8GIG2[e,~ ] = ~P2-SFIF2.~pI .
It follows from uniqueness that
[-,-]
is a functor. To see
that it is a 2-functor, observe that if v : ~ -- ~'
u : • ~ 9'
and
are modifications, then
5.11) vP 2.SFIF2.uP I : ~P2-SFIF2-~p I ~ ~'P2.SFIF2-~'P2 is a modification and hence corresponds to a natural transformation (5.±2)
[u,v]
:
[~,~] ~ [~',~']
Since this is natural, we get a 2-functor with codomain 2-Cat/AI×A 2 Compatibility with composition means that, given ~i : Fi ~ F~l ' i = !,2,3 , the diagrams
108
1,5
[FI'F2]
9
AX2 [F2'F3]
> [~I,F 3]
I
×
[~I'F2 ] A 2 [F2'~3]
[~I,~3 ]
> [Fi,F~] (5.i3)
[FI'~2]
I [El'F2]
[FI'F2]
AX2 [F~'F3]
~2 [~2'F3]
A~ [F2'F3]
o
...............o.
>
IF i
3]
commute. This follows by inserting 2-cells into the diagram defining composition inserted
in part i) in the same way that they are
into the basic diagram
Naturality
in defining
[~,~]
in all variables means, given a commutative
diagram of 2-functors,
quasi-natural
transformations
and
modifications S.
(5.14)
i=l, 2,
V • 1
A'
~
at
i
G'l
109
1,5
I0
then the diagram
[~i 'q°2] s
v
!
[s I , 2 ] I
[G I ,G 2 ]
A l x A2
(5. 15)
(V i ,W ,V2)
~VixV 2
EFI,~I ~
r ~ 21.1 '!
commutes. Remarks.
This
(V i ,W ,V2)
...E i' ;I
"--------
is easily established.
The construction
to give a 3-functor
in part
defined
°P[°P(2-Cat)
ii) above can be generalized
on 3-comma
]3x[ (2-Cat) ,r~]3 ~
categories
(2-Cat,PixP 2)
where a pair of morphisms Vi
AI
v 2
>
A~
A2
give rise to a diagram
1t0
(cf. 1,2.7) ,
-) A~
1,5
11
[F ,F2]
F1
which
induces
a morphism
(in the imprecisely
described
comma
category)
(V I ,~I ;V2 ,e2 )
IFI,~2]
>
[F~,F~]
(5.17
AlXa 2
-
Vl×V 2
and 2-cells
and 3-cells
and 3-cells
"over"
1,5.4. of components expressed
>
give rise to corresponding
their components
Explicit
AIxA ~
formulas.
as follows:
111
in the place of
Using
in the proof of 1,5.2,
2-cells VI×V 2
the description
these operations
in terms can be
c~
~j
I t~
t~
4~
t~
t~
~4
f
t~
~×
~
v
v
v
I
j~
t~
v
~J v
t~
1,5
c)
[~,~]
A morphism
in
13
: [FI,F2] [FI,F2]
[GI,G 2 ]
as
in a)
is taken
into
Gif 1
GiA I
>
G1A~
"~
FIA ~
Glf I FIA i
........
GIA i
Fif i
~
I* F2A 2
-
G2A 2
..........G2f2
[-,-]
being
[~',~']
'~
a functor means
: [GI,G2]
is o b v i o u s
another
~
[HI,H2]
that
It follows, the e x p l i c i t
G~A! z z
, then = [~,,~,~]
[~,~]
~ : Ai ~ [GI'Fi] either
>
if
from the e x p l i c i t
w a y to e x p r e s s
to a functor
_ ~ @2 r2
G2A ~
[~',~'][~,~]
which
]
G2A 2
F2f2
GIA ~
from
formulas,
formula.
; namely, and
the u n i v e r s a l that
~
Note
that
~ : GI ~ FI to
there
is
corresponds
~ : A 2 ~ [F2,G 2]
mapping
the d i a g r a m
113
,
property
or from
1,5
14
x [Fi,F2 ] A2 × [F2,G2 ] [G1,Fi ] A1
;xlx~)
lo [G i ,G 2 ] commutes. d)
[u,v]
: [e,~] ~ [~',~']
whose component at
is the natural transformation
(h:FIA i -- F2A2)
in
[FI,F2]
)
GIA I
)
G2A2
morphism i GIA I
G2A2
in
[Gi,G2]
[-,-]
1
being a 2-functor means that
[u',v']-[u,v]
=
[u'-u,v'-v]
[u',v']
=
[uu',v'v]
[u,v]
l
when defined. e) A morphism
(Vi,~I~V2,~ 2) : [FI,F2] in
[FI,F 2]
!
as in a) is taken into
114
!
> [FI,F 2 ]
is the
1,5
15
F[Vlf I > F~VIA 1
FIVIA 1
(%ol)~/~[ (%ol)A~
(~1)All
F~(Vifi)
FIA1
Fifi'> eiA~
(%o2)A~h' (%ol)A~
= (~2)A2h (~I) A:I F2A 2
e~(v2A2) .e~(v2f2~>F~(v2A~)
P2f2
(%o2) f
(%O2)A21 F~V2A 2
> q(V~A~)
F~(ViA i)
~
%o2)A~
~v2f 2'> F{v2A~
1,5.5. Definition. Given pairs of 2-functors FI
a 2-functor formation over
VIXV 2
F2
~1
~
~
(
41
> ~'(
A2
~,
! ! T : [FI,F2] ~ [FI,F2]
(resp., quasi-natural trans-
%O : T ~ T' ~ resp., modification
s : %O - %O' )
(resp., %OI×%O2 ; resp., sl×s 2 ) if [Fi ,F2]
T
Vi×V 2
115
~
[~I'F~]
is
1,5
commutes
16
(resp. ' ~l[D''P'2}9 = ~ I x ~ 2 { P 1 , P 2 } ; resp. ,
{v~. ,v~ }s = ~l×s2 {v 1,v 2 } . ) . The
2-functors,
modifications and 3 - c e l l s 3-category
which
of the
quasi-natural
are over s o m e t h i n g "full"
of the c o m m a
(i.e., all
3-category
x : 2-Catx2-Cat is c a r t e s i a n [Fl,P2]
product,
determined
~ AS×& 2 . Denote
By 1 , 2 . 9 ,
since
(2-Cat,x)
~ 2-Catx2-Cat
1,5.6.
transformations,
this
are the l-cells,
1,2, a n d 3-cells)
(2-Cat,x)
2-cells, sub
, where
~ 2-Cat by objects
sub 3 - c a t e g o r y
2-Cat has enriched
Proposition.
and
pullbaeks,
of the by
form
[_,_](2-Cat,x)
the projection
is a 3 - f i b r a t i o n .
i) The p r o j e c t i o n
[-,-] (2-Cat,×) ~ 2 - C a t × 2 - C a t is a 3 - f i b r a t i o n is c l e a v a g e ii)
[-,-] (2-Cat,×)
in
(2-Cat, x)
preserving. In p a r t i c u l a r ,
VixV 2 , there AixA 2
and the i n c l u s i o n
is a u n i q u e
given T'
T
: [FI,F2]
: [Fi,F2]
~ {F~Vl,F~V2]
such that the d i a g r a m
[F 1 ,F 2 ]
[vlv1,F~v 21
>
commutes.
1t6
! t ~ [FI,F2]
over over
1,5
iii) between
Similarly,
natural
and n a t u r a l where
T
Proof.
i)
follows
there
17
is a b i j e c t i v e
transformations
transformations
, T 1 , T' (2-Cat,×)
~ : T ~ T1
e'
T1
, and
correspondence
: T' ~ T~
over over
Vi×V 2 AIxA 2 ,
are as in ii).
is a 3 - f i b r a t i o n
from the o b s e r v a t i o n
(which
via p u l l b a c k s ,
is i m m e d i a t e
so this
from
(5.1))
that (V 1 ,1 ,V2)
[~Ivi,~v 2]
t
al×A2
is a p u l l b a c k
..... V l x v 2
in
>
[~I,F~]
)
I
Alxa~
2-Cat
ii) and iii) just r e p a r e n t h e s i s e s
are special T ~ ire.,
T(F i(A i) ~ F 2(A 2))
cases
of i). E x p l i c i t l y ,
T'
if
= F l(v i(A i)) ~ F ~ ( V 2(A 2))
then T' (FI(A I) ~ F2(A2)) We note
= F ~ V I ( A I) ~ F ~ V z ( A 2)
that any q u a s i - n a t u r a l
transformation
over
VIXV 2
is a u t o m a t i c a l l y
natural
over
VI×V 2
is a u t o m a t i c a l l y
the i d e n t i t y
Either using
(2.18)
2-functor
by identifying or d i r e c t l y
(V i ,I ,V2)
and that a m o d i f i c a t i o n
(Vl,i,V2)
one can d e d u c e
when
Vi
and
117
~ : T ~ TI
V2
(cf.,
1,5.2,
ii).)
= (Vl,id~V2,id) what happens
to a
are v a r i e d b y
and
1,5 quasi-natural W
differs
transformations.
18
The corresponding
behavior
in a crucial way from the analogous behavior of
comma objects. This is, in fact, equivalent composition of quasi-natural
transformations
to the failure of to be 2-functorial.
For brevity we write
(5.18)
W, = (i,W,i)
: [FI,F2] -- [WFI,WF2]
1,5.7. Proposition. Given 2-functors and quasi-natural
transformations
V,
A[
~i[
W Ai
i = 1,2
i ) Wt
V! 1
then i) the diagram
[Fi~ I ,F2~ 2 ] [FIV I,F2V 2]
~>
[F1,F 2 ] commutes,
in
and
ii) there is a diagram
118
[El IV' ,F2 2V']
1,5
[Fi,F2 ]
...... W~
[WtFI'W'F 2 ] over
19
AixA 2 , where
>
/
[,Fi,i ] ~.
[WFi,WF2 ]
~
[WFI,W'F 2 ]
is a Cat-natural
transformation.
Proof. i) is clear. To prove ii) observe that are induced respectively by
WOFIF2
and
diagram reflects the composition
~O
f : FIA i ~ F2A 2
[FI,F2]
W.
WtOFiF2
and
W.t
and this
. HDwever, it only FiF 2 seems possible to describe it in terms of components. Let be an object in Wf
WFIA i
~.
. Then, in the diagram
WF2A 2
(5 .i9)
~FIAi1
~f
WtFiA1 .......... W'f
~F2A2
the clockwise composition
is
clockwise composition
[~F2,1]W~(f)
to be the morphism
in
Explicit calculations given a i-cell [FI,F2]
is
•~I~F2A2
[i,~F2]W.(f)
[WFI,W'F2]
while the counter-
. We define
represented by
show that this is Cat-natural.
(hi,~,h 2) : (Ai,f,A 2) ~ (A~,f',A~)
(~.)f (i,~f,l) E.g., in
, then the equation
(~.) ft-[~F 2,1]W~(h i,~,h2) = [I,~F2]W .(hi,~,h2) • (~.) f is equivalent to the commutativity
119
of the cube
1,5
20
WFIA 1
%Pf
WFIA l
WtFIA 1
W'F2A 2
which
follows from 1,2.4, QN1 and QN3. C a t - n a t u r a l i t y follows
s i m i l a r l y from 1,2.4, QN1. The d i a g r a m s d e s c r i b e d
in this
p r o p o s i t i o n have m a n y c o m p a t i b i l i t y p r o p e r t i e s whose e l u c i d a tion we leave for a later paper. There
is another point of view w h i c h is v e r y useful
in describing
further p r o p e r t i e s of 2-comma categories;
&l ~ [ F I ' F 2 ] can be r e g a r d e d as a 1-cell In particular,
if
~ A2
in Spans(2-Cat)
F : A ~ ~ , then
in the m u l t i p l i c a t i v e c a t e g o r y
1,5.8. Proposition.
Let
namely,
(cf., 1,3.4
[F,F] ~ AxA
is an object
[Spans(2-Cat) ](A,A)
F
l
: A
1
~ ~ , i = 1,2
be
2-functors i) tion
{P1,P2}
: [F1,F2] ~ AI×A 2
is a s p l i t - n o r m a l
2-fibra-
(cf., 1,2.9). ii)
[FI,F I] 2 A i
is a m o n o i d in
120
Spans(2-Cat) (AI,A I)
1,5
and if
W : ~ - ~'
then
W,
21
: [FI,FI] -- [MFI,MF I]
is a
monoid homomorphism. iii) is a left
A i ~ [FI,F2]
[FI,Fi]-right
and the bimodule
~ A2 [F2,F2]-bimodule
structure
induced on
in
Spans(2-Cat) (Ai,X 2)
[FI,F2]
by the change
of monoids
coincides with that given by the bifibration iv) If formations,
~i : Fi ~ F!I ' i = 1,2
then
[FI,~2]
module h o m o m o r p h i s m s cleavage W,
preserving
i) In [CCS],
[FI,F2] O ~
(AI×A2) O
are quasi-natural
[@I,F2]
respectively, respectively.
: [FI,F2] ~ [WFI,WF2]
Proof.
and
structure. trans-
are left and right
and hence cleavage
and op-
Similarly,
is a bimodule homomorphism.
§6, Example is the
I, it is pointed out that
(1,o)-bifibration
corresponding
to the functor ~(Fi(-) ,F2(-)) and as a 2-category of [CCS], sense of
: Alx~ 2 ~ Cat
it is given by the fundamental
construction
§5. We claim here that it is a 2-bifibration (1,2.9) . The split cleavage
and opcleavage
by taking as cartesian and opcartesian morphisms those morphisms
of the forms
121
in
in the
are given [FI,F 2]
1,5
(5.20)
22
Fim
i
FIA i
> FiA ~
h' (Fim)I
F2A2
cartesian
...........
FIA i
j h'
1
'~
and
F2A~
'> FiA i
h~
opcartesian
F2A2
F2n
l(P2n)h
> F2A~
respectively. Treating only the cartesian case, this defines an object function iLl
: I (A1,P1) I "
taking the object
I ([F1,F2])~I
(m,h') to the indicated cartesian morphism.
It is easily verified that this extends to a 2-functor L satisfying
SL = id , where S = {4,[FI,F2]
} : [Fi,F2
~ (~i,Pi) ,
and one must only verify that S Ca--~-qtL . Consider the diagram
Fim FiA i
~> Fi~~ Flm
FIA i
Fia1 h'
h' (Flm)
F2A 2
F2n
1'
V~2t F2A ~
-,,
> F2A~
122
'
1,5
where
(n,~,m)
mapped
into
L(m,h')
Commutativity which
: h ~ h'
23
is r e g a r d e d
b y the m a p s
of the c u b e
shows t h a t t h e r e
shows
as an o b j e c t
(t,y,s)
that
and
between
[FI,F2]~
(t',y',s')
t = t~n
is a b i j e c t i o n
in
and
¥ = ¥' ~
such 1 - c e l l s
and
1-cells [F1,F2]
b)
¥W,T
(resp.,
[F2,Fi]AI[FI,F1]
¥'W,T,)
is c o m p a t i b l e
° > [F2,F 1]
with units
and
associativity. c) AI×A 2
If,
furthermore,
(resp.,
T~
T 1 : [WFI,G ] ~ [W'WFI,H]
: [G,WFI]
~ [H2,W'WFI]
is over
) then
YWtW,TI T = YW,,TI [ ] ¥ W , T
(resp. 'Yw'w,TIT' ii) a) T ¥W,T = id (resp.,
or resp.,
opcleavage) b)
(resp.,
T'
If there )
quasi-natural
= [~,WFi]'W
c) embedding)
T'
YW,T"
) is a h o m o m o r p h i s m = id ) if and o n l y
exists
an
N
transformation (resp.,
with
The
if
F 2 = FiN
if and o n l y ~ : WF 2 ~ G
e : G ~ WF 2
T
is c l e a v a g e
then
if there
T is a
such that
such that
. ).
If further, then
~
Fi
is fully
is u n i q u e
up
unique). Note:
(i.e.,
preserving.
is a h o m o m o r p h i s m
T = [WFI,~]-W . T'
(resp.,
= ¥W',T~ ~ YW,T' )
special
case of functors
125
faithful
(resp.,
to an i s o m o r p h i s m
a full (resp.,
I ,5 ~.
26
[~,F]
FJ, ~
and
T ...>
~
[W,G]
/
(5.21)
leads to T being a homomorphism if and only if it is of the form T = [W,e]-W, where ~ : WF ~ G . This (and its dual) will be specialized further in 1,5.10 to the cases where either W
or G
is the identity.
Proof: i) Consider the diagram (FI)*Ax$
[Fi,F1]Axi[F$,F2] M
2x
a~'A1 [ F1, F 2 ]
->
[F1,F 2 ]
w. x~
I~T i
T
[w~'i ,~l] a\ [wF1, a]
....
2
A--x [WFi,G]
lh
where M and
~
and M'
are given as in the proof of 1,5.8 iii)
is the natural transformation for T
as in 1,3.5.
We want to factor this through ¥ as indicated. The hypotheses
126
1,5
27
are needed to make this work. The procedure in 1,3.5 suggests the following: given an object
((~IAl m~ ~IAi),(~IA i h ) F2A2)) ~ [F1,FI]A~[FI,F2] , since
FI
is full we can write
then apply T
m =Fln
for some
to the cartesian morphism for
lhm
lh
=
n
and
(n,h) ~ i.e.,
{
IT(hm)T(h)/
\
~YW'T) r e , h / ~
(5.22) /
The result is not a cartesian morphism in general. The domain of the component of the comparison natural transformation from it to the cartesian morphism for
(n,T(h)) is the
morphism
IAl
i
>
T(hm) [
IAi
(5.23)
IT(h)W(m) (YW,T)m,h/~k~~
GA 2 in
[WFI,G] from
>
T("o") (m,h)
to
(" o") (W,A~T) (m,h)
This does not depend on the choice of satisfies
n
Fl(n') = F1(n) = m , then~since
full, there is a 2-cell
~ : n ~ n'
127
GA 2
with
since if F
n'
also
is locally
Fi(~) = id .
1,5
The result Ij2.5
follows by putting
and the fact that Naturality
means
T
28
this
2-cell
in (5.22)
is a 2-functor
over
and using
AixA 2 .
that given a pair of morphisms
(g' ,T,g) :(A~,m,A I) -- (A~,m,i i) (g,~,f) :(Ai,h,A 2) ~
(AI,h,A2)
in
[F1,F i ]
in
[Fl,F 2 ]
then (YM,T)~,~]
T(o ~ T) = [T(~) [] M(T)] ~
where w e have written This
follows by expanding
(since using
T(a)
F
is locally
for the 2-cell (5.22)
full)
(¥M,T)m,n in
T(g,~,f)
into a cube with
as a 2-cell
(5.24)
T = F(T')
in the top face and
1,2.5 again. i) b) Compatibility
with units means
that
(¥W,T) (jF I A×I [FI'F 2 ]) = id • This
, etc.
follows
Compatibility
since
T
is a functor,
with associativity
128
so
means
(¥W,T)FIAI,h
(5.25) = id .
that the diagram
1,5
29 ix~
°
[Fi,FI]~I[FI,F2]
.>.
[FI,F 2]
w. T
W* ~x W. x T
[WFI 'WF1 ] ~1[ WF1 'WF1] ~1x[WE I ,G]
ixo
)[WF i ,WF i ] ~i [WEI'G]
[~l 'WFal~[WF1,Gl commutes.
This follows
FIA ~
since applying
m'
>
T m
FiA 1
to a composition
> FiA i
:ram, F2A 2
as in (5.22),
i
(5.27)
>
F2A 2
........
1
>
F2A 2
shows that Ymm ~,h = Ym,n ~ ¥m I ,hm ;
i.e.
which
Ymm',h = [¥m,h (Win') ]'¥m',hm is what is needed when
¥
is interpreted
i) c) This follows by applying
129
Ti
as in (5.23).
to the composition
1,5 I
W(m)
> WqA 1
T(hm)1
>
'
>
GA 2
of the second gives
TI((Yw,T)m, h)
(?W,,TI)Wm,Th , while
morphism. Conversely, suppose T F 2 = FIN , the identity map of regarded as an object of ~A 2
=
T
Ti
T 1 of the composi-
>
[WFI,~]-W.
is a homo-
is a homomorphism. Since F2A 2
into itself can be
[FI,F2] . Set
(idF2A2:FINA2 ~ F2A 2)
f • A 2 ~ A~ , ~f
FINA (5.31)
and
(¥WtW,TIT)m,h .
ii) It is clear from 1,5.8 that
and for
(5.29)
,,,
T i of the first square gives
tion gives
WFIA1
IT(h)W(m)
GA 2 since
30
:
(5.3o)
WF2A 2 ~ GA 2
is the 2-cell in the diagram
FI A
FINf
T
->
WF2A 2 1
1
21
WF2 f
WF2A I
= 2
F2A2 Since
T
F2f
>
F2A~ /
is a 2-functor, ~
Alternatively, let T
GA2/
Gf ~
~/A2
is a quasi-natural transformation.
be the unique 2-functor over
(by 1,5.5) making the diagram
130
A2xA2
1,5
[F 2 ,F2 ] =[FIN,F2 ]
31
(N,I ,i)
>
[F I ,F2]
I
~,t
(5.32)
T
I
[~2,GI=[~!N,~I commute
and then
corresponding
>
(N,1,1)
~
[~I,Sj
is the quasi-natural
transformation
to 3F 2 = A2
This formula not
T
> [F2,F2] always
defines
is a homomorphism.
When
for objects
(¥W,T)h,idF2A2
for morphisms
Alternatively, vity of
(5.34), where
is a homomorphism,
or
then
o W,
~ F2A2)
~ [FI'FI]
= ~A 2W(h) follows
the equation the regions
II commutes
by 1,5.4 c) , I I I
IV commutes
by the definition
does not hold
, whether
in
~i [F!'F2]
= id , we get that
= T(idF2A2)W(h)
The equation
~ : WF 2 ~ G
as the first component
(h:FIAl ~ FINA2'idF2A2:FINA2
T(h)
(5.33)
by considering
(h:FIA i ~ F2A 2) £ [FI,F2]
Since
a
T
T = [WFI,e] follows
>[WF2,G ]
in general)
from
labeled
from the commutatiI commute
by 1,5.3
by the definition
~ , and
V commutes
131
(5.24).
follows
commutes of
= ([WFI,e],W .) (h)
of
i) , T
,
(the only thing which
because
T
is a homomorphism.
C~ ¢'J
H
h~
[F I ,F2 ]
W.
•
X
[F I ,F2 ] A2A 2
W*A2A 2
X [WF1 'WF2 ] A2 A2
%
[WF 1 ,WF 2 ]
i
1
, 1×iF2
~v
->
[WF 1 ,G]
/
-> [FI,F 2 ]
>
[wF i ,~1
lo
× (1 ,1 ,N)~ (N,i ,1) ) [WF I ,WF I ] al[ WF I ,G]
III
x [FI, F 2 ] (1, I",N) ~ (N,1,1) > [FI,FI] AI
[F 1 ,F 2 ]
/
/
> [FI,F2] A2[F2,F2 ]
,
l
IF1 'FIN] ~2 [FIN 'F2 ]
~*~2T
[WF1 'WFIN] A% [WFIN'G]
->
ix$ > [WF1 'WF2 ] Xx2[WF2 ,G] II [WF1, ~]
(5.34)
e4 ¢o
1,5
It is c l e a r way that
F2A 2
faithful of
and
F2A 2
Then
~
depends
is r e p r e s e n t e d
if
is
that
F1
applied
= (~A2 ,SA2,~'A2)
transformations (since
this a m b i g u i t y
be u s e d 1,5.7.
so
in 1,7. O t h e r c a s e s
follow
We o m i t c o n s i d e r a t i o n
1,5.10.
Corollary:
i) Let
F,G
(resp., T'
~
four s p e c i a l c a s e s
dealt with more carefully
~xA
= id.)
: a ~ ~
: [F,~] ~
= i.e., F1
, then the
identity map
If
FI
m
- A 1 ~ A~
the c o m p o n e n t s between
with
on the
is f u l l y
isomorphism
provides
constructed
WFI(m)
N
F I A ~ . If
s : ~' ~ ,"
cannot occur,
T h e r e are
o n l y on
to a u n i q u e
an i s o m o r p h i c m o d i f i c a t i o n
respectively
as
FIA ~ = FIA 1 = F 2 A 2
T(m,id,1)
natural
33
for
the q u a s i -
N A 2 = A~
or
A~'
is an e m b e d d i n g ,
is u n i q u e . of this t h e o r e m w h i c h w i l l from these
of p a r t c) h e r e
together with since
it w i l l be
there.
and let [G,~]
T
over
: [~,F] ~
[~,G]
be o v e r
A x ~ ). T h e n there
is a
diagram
[,z,s],~[,z,F]
'~. [~,~]
[r,~]~[~,,"*]
°
>
[~,~]
[,z,,z]~[,,,c-.]
o > [~'Q]
[c-,,~]~[,,,~]
o '>
[~,~]
and
YT
(resp.,
y¢, ) is the
a unique quasi-natural T = [~,~]
(resp.,
~'
identity
transformation : G ~ F
with
133
if and o n l y ~ : E -- G
T' = [~',~]
if t h e r e with
).
is
I ,5
ii) T
: [A,U]
over
Let ~
~x~
F
: ~ ~
[F,~]
) . Then
be
~
and
over
there
34
U
: ~ ~
X×~
is
A
(resp.,
and T'
let
: IF,Z]
~
[A,U]
a diagram
X
F, xT I
¥
[F,F]x[F,~] A
and
is
with
T =
Proof: and
o
YF,T
there
a
a unique
i)
~5 , w h i c h
FI =
embedding, For N
and
G
= A,
W = U
the = U
we
found
later
1,5.7
is g i v e n
It
is
. For
[A,U]x[U,U] ~
o
identity
if
and
: A ~ UF is
with
certainly
G = G, let
second
s
W =
full,
: FU =
[A,U]
if ~
~5
[~,U]oU,
locally
and
X,
F 2 = U,
let
'>
only
Z
FI = part,
T'
Tt
N = F
Fi =
full .
G = ~,
).
~,
F 2 = F,
N = F.
of
examples
shall
in t h i s by
not
work.
taking
corresponding
easily
¥U'
transformation
part, the
T'~U,
Examples.
A number and
~
F 2 = F,
first
and
1,5.il.
the
quasi-natural
Let
ii)
the
) is
(resp.,
= F
Denote
¥~-,T"
[F,S]oF,
full
[CCS]
~esp. ,
> [F,~]
(resp.,
W
in
T
seen
is a q u a s i - n a t u r a l
that
of
2-comma
repeat
categories
them
here.
A particular
case
a pair
of
2-functors
[ rBn , ~ n ]
transformation
134
Many
objects from
= ~(B,C) between
which in
!
can
be
more
found
will
be
illustrates
a 2-category
~
by
and,
if
2-functors
~
: F ~ G
from
.
1,5
to
~'
, t h e n the
a.
in 1,5.7
35
is the
1,2.4.
135
same
as the
~BC
in
1,6
1,6. A d j o i n t m o r p h i s m s
in
I
2-categories.
In this section we discuss p r o p e r t i e s of adjoint l-cells in a 2 - c a t e g o r y and, briefly,
in a bicategory.
equational d e s c r i p t i o n of adjoints
The usual
is taken as the definition.
It is impossible to state e v e r y t h i n g that follows from this definition,
but we have tried to organize one class of p h e n o m e n a
around the n o t i o n of the c a t e g o r y of adjoint squares category
in a 2-
A, and using this, another class around the notion of
Kan extensions along a 1-cell in a 2-category.
In this section
we ignore q u e s t i o n s of existence of Kan extensions
(since
nothing can be said about this in a general Z-category)
and
c o n c e n t r a t e on d e s c r i b i n g the r e l a t i o n s they satisfy w h e n they exist.
It should be e m p h a s i z e d that any such thing
formal". What
is interesting
is "purely
is how much a c t u a l l y fits that
description. As applications, we discuss p r e s e r v a t i o n of Kan ext e n s i o n s and,
in particular,
of colimits,
including the "formal
c r i t e r i o n for the e x i s t e n c e of an adjoint" of Dubuc as the usual
[9 ] as well
interchange 2-cells b e t w e e n limits and b e t w e e n
limits and colimits. We also b r i e f l y discuss and "dual" Kan e x t e n s i o n s
"final" m o r p h i s m s
in Cat.
1,6.1 Definition: i) Let A be a 2-category.
A pair of m o r p h i s m s
f A(
>B u
is called adjoint, w r i t t e n e: fu
} B,
l u, if there exist 2-cells
f 9: A
~>uf
such that
136
1,6
2
sf-fN = f (6.1) = U
Ug'~U
ii) Let ~ be a b i c a t e g o r y . above
is c a l l e d a d j o i n t
A p a i r of m o r p h i s m s
if t h e r e e x i s t e: fu
>I B
q: I A
>uf
(f,u)
as
2-cells
such that the c o m p o s i t i o n s -I f
-I
r
~>fI A
f~.)f(uf)
~
ef > (f~)f
:~ )f
>IBf
(6.2) e -i
r
u
) IA u
are
~U > ( u f ) u
a
~ u(fu)
us > U i B "
>u
identity 2-cells 1,6.2 E x a m p l e s I) A d j o i n t m o r p h i s m s
simple
fashion.
opfop
denote
opAop
the c o r r e s p o n d i n g
f
b) °Pu
....o.p f
c) u Op
4 fop
2) In a
only
..~..G
in a a n d if
morphisms
in
then the f o l l o w i n g
in a v e r y
opf, opt,
fop
~op
and and
are e q u i v a l e n t :
and
jlOPuOP. ~category ~: G
~, ~F
Cat-natural
the r e q u i r e d
transformations
are a d j o i n t ,
if t h e r e are m o d i f i c a t i o n s
satisfying
dualize
lu
d) o p f o p
~: F
If f is a m o r p h i s m
, respectively, a)
in a 2 - c a t e g o r y
~:
equations.
137
~ This
-
~ .......4~ =~ G
and
, if a n d ~: F
is e q u i v a l e n t
to
1,6
requiring
that
expressing
these
"naturally"~ 3)
the
4 ~B
i.e.,
as c o m p o n e n t s
to r e q u i r i n g
expressing
so that,
given
that
~
e: e ~
and
these
2-cells
qB' c a n be c h o s e n
are a d j o i n t ,
for all
the
of m o d i f i c a t i o n s .
)G
equations.
that
A and
Fun(~,A) , q u a s i - n a t u r a l
~: G ...... > F
the r e q u i r e d
in
e B and
are m o d i f i c a t i o n s
2-cells
chosen
~B
adjunctions,
and
satisfying valent
B,
In a 2 - c a t e g o r y
~: F .....~. G if t h e r e
for all
3
In this
....|~ ..
adjunctions,
, if and o n l y
q: F
case,
B, ~B-----~
transformations
this
~B
eB
%~ is e q u i -
in A and
and
qB
that
c a n be
a 2-cell f
f' in
~, t h e y
satisfy
(~C~f,.~f,~B) . [ (F~) qB ] = qc(FN)
(6.3) [e c(G#) ] - ( ~ C ~ f . e f ~ B ) = (See
1,2.4,
MQN.)
Similar Bifun
formulas
hold
for a d j u n c t i o n s
in a b i c a t e g o r y
(~,~') . 4)
In the
2-functors
F:
Cat-adjoint,
= F
existence
A
2-category >~,
U:
and written
transformations eF-Fq
(G#)eB
and
e: F U ~ Ue-qU
2-Cat ~ •
(see >A
are
F
Cat ~ U
~ ~
and
A ~ this
isomorphism
138
adjoint
if t h e r e
q:
= U. As u s u a l ,
of a C a t - n a t u r a l
1,2.3) , a p a i r
>UF
of
(also c a l l e d
are C a t - n a t u r a l satisfying
is e q u i v a l e n t
to the
1,6
4
~(F(-) ,-) ~ ~%(-,U(-)) : A°p × ~
5) In particular, 4: ~
)A m, where
Cat-adjoints
Cat-limD ' Cat limit
' ~ ~
instance,
(See,e.g., Dubuc
to a constant embedding
[9 ]-) More
4: A
are called Cat2-1imits
~ X , and
written
Cat2-1im ~
For
if and only if it is preserved
functors
where ~ is a small 2-category, Cat2-colimits,
are called Cat-
Cat -l!m
is a Cat-limit
Cat-adjoints
imbedding
written
by the Cat-representable generally,
to a constant
~ is a small category,
limits and Cat-colimits,
An ordinary
~ Cat
Ca------~~
Ca~{
the Cat2-1imit
Cat-lim~
of a diagram of the form F
(6.4)
A
B
f'
~
u
with
f---~u
and
f'
iii) A d j o i n t s
P r o o f . i) L e t adjunction
~t:
A
respectively.
with
from the commutativity
u'fu' ~ u'f~u'
by
square
fabled
is a f u n c t o r
2-functors.
f----4 u
Q: A ........~ u f
and
f
c':
fut
and
~u'
ut.
)u
u's-~'u
) B,
(6.5) as
its
inverse.
The proof
.~ ufu'
ua'
) u ' f u f u ' ..... u ' f u ~ '
I commutes,
.> u ' f u
..... for
>u
)u'
instance,because
This
fact
is a p p l i e d
composition to the
composition ufu'
A
u
utf
the
follows
o f the d i a g r a m
of two variables.
In a b i c a t e g o r y ,
with
Then
u'fu t The
luu t.
Suppose
e: fu ' ) B ,
nu'
f'f '
are p r e s e r v e d
2-cells
isomorphism
u'
then
A be a 2-category.
)u'f
>C
u v
I u',
u~1-~u: is a n
5
isomorphism
composition
140
u'
..>u
is g i v e n
b y the
1,6
6
_i u'
•
~
)iAu ,
~u' >(uf)u'
•
~ 'tu(fu')
•
ue'
>
r
uI B,
) u
(6.6)
w i t h an a n a l o g o u s formula for the inverse. The same d i a g r a m as above, c o n s i d e r a b l y e x p a n d e d to take care of the a s s o c i a t i v i t i e s and units, -that c ~ p o s i t i o n
provides the proof via the same a r g u m e n t
is a functor o~ two variables.
ii) Let X be a 2-category. e: fu st:
then define
If the a d j u n c t i o n 2-cells are
9B
q: A
f'ut---#C
~': B
e~
and
~" = ( f ' f u u '
)uf )utf t
~" to be the c o m p o s i t i o n s
f'eu'
e'
~f'u t •
>C) (6.7)
~" =
(A
~
"~ uf
These describe an a d j u n c t i o n (uu'e").(~"uu')
uu t .~
= id
quu'
holds,
uD'f
~ uu'f'f)
f'f----4uu'. for instance, uqt fuu t
% ufuu'
The equation since the diagram
> uu' ft fuu t
I
uu
UNU t
t
)uu' f'u'
uu
commutes.
Again the crucial
functor of two variables.
ingredient
uut fteut
~
is that c o m p o s i t i o n
In the b i c a t e g o r y case,
appropriate
instances of the isomoL~phisms ~,~, and r must be inserted, As these results suggest,
as in i).
almost any equational p r o p e r t y
of adjoints carries over a u t o m a t i c a l l y to 2-categories,
141
is a
and w i t h
1,6
sufficient
care to bicategories.
results which is the case
iii)
this is not true for
in an essential way,
in the following proposition,
2-category
ii)
However,
use the Yoneda embedding
1,6.4 Proposition.
i) f --~ u
7
The following
as
and its consequences.
are equivalent,
for a
A: in
A(-,f)
A, 4 ~(-,u)
in
Cat AOp ,
l A(f,-)
in
Cat A.
A(u,-)
Proof. The implications from 1,6.3,iii). since Yoneda
i) --ii)
Conversely,
and iii)
given
follow immediately
ii) , for instance,
then
is full~ the m o d i f i c a t i o n s ~: A(-,f) oA(-,u) ~: A(-,A)
o) A(-,B)
) A(-,u) oA(-,f)
are induced by 2-cells = (~B)id B: fu = ~ B q = (~A) idA: A ---~uf The adjunction
identity Za(-,f),a(-,f)q
applied
to
idA: A-
(~A) f =
(~'A)idAf, so
;A
= id (sA) f'f(qA )id A = id.
yields
sf-fq = id. The other
By
(2.8) ,
identity is derived
analogously. For a bicategory,
Yoneda
is not locally full and this
argument cannot be carried out. We have not attempted out what is true in this case, except that, clearly, ii) and iii).
142
to find i) implies
1,6
8
1,6.5 Remarks: C o n d i t i o n
(by 1,6.2, (I)) that
ii) means
for all C, the functors
A(C,A)~ A(C,f) X(C,u) are adjoint
in the usual
sense and that the a d j u n c t i o n natural
t r a n s f o r m a t i o n s are natural given
h: C
)A, k: C
) A(C,B)
in C. T h i s , i n turn, m e a n s that
)B, there are b i j e c t i o n s
A(C,B) (fh,k)
c A(C,A) (h,uk)
(6.8)
w h i c h are natural with respect to varying h and k by 2-cells and by c o m p o s i t i o n w i t h
c: C'
)C. The dual p r o p e r t i e s hold
for c o n d i t i o n iii) w h i c h gives b i j e c t i o n s A(A,C) (hu,k) for h: A
)C, k: B
z A(B,C) (h,kf)
%C.
1,6.6 Corollary. adjoint
morphisms
(6.9)
(Cf., Palmquist,
f~
~u
and
f'
[37])
lu'
and
Given m o r p h i s m s h and
k as indicated k
B
>B v
(6.10) A
=>A' h
there
is a c o m m u t a t i v e
square of b i j e c t i o n s
A(A, B' ) (f'h,kf)
-~ A(B,B') (f'hu,k)
(6.11) A(A,A' ) (h,u'kf)
-~ ~(B,A') (hu,uWk)
Proof. The b i j e c t i o n s are given by the preceeding remark. The h o r i z o n t a l ones can be d e c o m p o s e d by s t a n d a r d a r g u m e n t s b y
143
1,6
inserting
a column
A(B,B') (f'hu,kfu) in the m i d d l e A(u,A') respect right
9
. The
~ ~(B,A') (hu,u'kfu)
left side m a p s
and a c o m m u t a t i v e to c o m p o s i t i o n
side b y
commutative
square
results
w i t h the m a p
~(B,B') (f'hu,ks)
square
to it b y X(u,B')
ks: kfu
)A. It m a p s
A(B,A') (hu,u'k~)
results by naturality
ing k b y the 2 - c e l l
by naturality
u: B
and
and
~k. H e n c e
with
respect
the o r i g i n a l
with to the and a
to v a r y square
commutes.
1,6.7.
Definition.
An a d j o i n t
p a i r of a d j o i n t m o r p h i s m s h a n d k as in 1,6.6,
and
f'
lu'
of a
and m o r p h i s m s
@12
@21 2-cells
have also been
|u
in A c o n s i s t s
together with a matrix
/~11
of c o m p a t i b l e
f
square
@22 arranged
in the p a t t e r n
studied by Palmquist
of 1,6.6.
[37] and M a r a n d a
(These
[36].)
T h u s we h a v e
I
~11:
fth--~kf
~12:
~21
h ~
~22: hu
~u'kf
fthu ~u'
and t h e s e are r e l a t e d b y the e q u a t i o n s ~11 = =
~12 = =
(e12 f) ° (f'h~)
=
(s'kf) - (f'e21) (6.12)
(e'kf) - (f'e22 f)- (f'hD)
(k~) - (~11 u)
= (e'ks) - (f'~21 u)
(6.13)
(s'k) • (f'~22)
144
1,6
@21 =
(ui~ll) " (D'h)
~22 =
(u'kc) • (u'~iiu) • (~'hu)
=
10
= (U'el2 f) " (q'hq) =
=
(6.14)
(#22 f) - (hD)
(6.15)
(u'ei 2) - (q'hu)
(uikc) • (@21u)
We shall call
such 2 - c e l l s
It f o l l o w s two a d j o i n t
directly
of e a c h other.
from t h e s e e q u a t i o n s
that
if one h a s
squares k
B
transposes
A
>B'
B
)
A
A t
k !
>B'
~A t
h
h'
and a p a i r of 2 - c e l l s
p: h---->h'
and
~ = k ........ >k' ..
then the
equations
v'~12
=
, .f,~
=
I .f1~u (6.16)
u'u-e22 u'~f'~21 are e q u i v a l e n t .
We c a l l
these equations
hold.
Adjoint treating
squares
relations
their properties
= ~ 2 - ~u !
.
= @21 N N and
between
adjoint morphisms.
in a l e c t u r e
quist
[unpublished],
[37]. The
p a i r of 2 - c e l l s
are a v e r y h a n d y b o o k e e p i n g
things were treated previously Benabou
v a compatible
at O b e r w o l f a c h by Maranda
and s u b s e q u e n t l y
following
theorem
145
devise
if
for
I first discussed in 1966.
Similar
[36] and, p r e s u m a b l y ,
by Dubuc
is the b a s i s
[ 9 ] and P a l m -
for the c a l c u l u s
of
11
1,6
adjoint
squares.
1,6.8 T h e o r e m - D e f i n i t i o n . of adjoint Fun(~
squares
Proof.
categories
Consider
the
double
following
%
adjoint
B
Ad-Fun(~).
squares
-~.
k !
B t
> B"
U "
'
> A'
h
domain
left and right
sides
A
cf.I,2.8)
object
111 "P121
The h o r i z o n t a l
B
of a c a t e g o r y
~22 k
identities
d e n o t e d by Ad-
V!
< ~21
and c o d o m a i n
The class
Ct
v
n
category,
category
(triple category,
C
Iu
A be a 2-category.
in A forms a double
. It is the u n d e r l y i n g
in double
Let
"
h'
and c o d o m a i n
of an adjoint
respectively,
while
are the top and bottom.
are r e p r e s e n t e d B
A
b y the
A
"~A
A
square
the v e r t i c a l
Horizontal
following
.) B
146
-~ A"
h
domain
and v e r t i c a l
adjoint h
are the
squares:
>A t
) A'
12
1,6 Horizontal composition
B
is given by k'k
,
~. B "
u"
~I
~21
*
A where
~2
~2~J~22 .......
h 'h
} A"
* ~12 = ( e ~ 2 e 1 2 ) ' ( f ' ' h ' q ' h u ) (6.17)
~21 * ~21 = (u"kt~tkf)" (~1~21) and ~
is defined in I~2.1, equations
(2.3) and (2.3)'. Vertical
composition is given by C
7' C'
If( ~11
gf
B
'
fl
u ~22 A
The 2-cells conjNqates.
ut
/ ~A
A
~II: f'-----~f and
~22: u
~u' are often called
T h e y determine not o n l y each other but also the
other two transposes,
~12:
f'u--}B
and
~21: A~
V e r t i c a l c o m p o s i t i o n shows that c o n j u g a t i o n
9u'f.
is c o m p a t i b l e w i t h
c o m p o s i t i o n of adjoints, while h o r i z o n t a l c o m p o s i t i o n shows that it is "functorial". The proof of 1,6.3 is a special case. N o w define a 3-cell of
~
b e t w e e n a pair of
adjoint squares of the form B
k
--~"
B'
B *
k'
~ B'
V t
A"
h
~, A'
A"
--> A' h'
149
1,6
(note
that a)
b)
the
four
objects
conjugate
2-cells
15
are
the
same)
911:
g .
).f,
922:
u.
~11:
g'
2f',
92~:
u' ....... >v'
2-cells
N: h
B
of
~v
. >h',
u: k - - > k '
with
the c o m p o s e d
are c o m p a t i b l e
such
that
adjoint
k'
"~B
B
to c o n s i s t
p and
v
squares
>B'
V !
A
3A
'A
) A'
h'
and k
S t
> B'
-~ B'
B
f'
V t
u t
\'21 A'
At
h
If one c o n c e n t r a t e s patibility From
this
obvious
says
horizontal
~
on the
a cube that
of c o n j u g a t e s
to
1,6.9
Ad-Fun
as
like
A
component,
the one
the t h i r d ones,
and weak
category
in 1,4.1,
composition, given
that
com-
QF23
commutes.
besides
the
by horizontal
composition
and
then
this
of
2-cells
third
makes
structure
is d i s c r e t e .
Proposition.
square
(2,2)
and v e r t i c a l
into a t r i p l e
restricted
adjoint
that
it is c l e a r
composition Ad-Fun
just
The
following
in 1,6.7
150
are e q u i v a l e n t
for an
I ,6
16
i) The square is an isomorphism in
Ad-Fun(A)
with
respect to h o r i z o n t a l composition. ii) h,k and
011
iii) h,k and In particular, isomorphism
Proof.
are isomorphisms.
022
are isomorphisms.
if h and k are isomorphisms,
if and o n l y if
022
then
01 i
is an
is an isomorphism.
It is clear from the d e f i n i t i o n of c o m p o s i t i o n that in
the diagram k
k -I
B
B t
11
B
012
G'11
u
f'
A
u
A'
A
h a candidate
G'12
u'
h-1
for the inverse to the first square must have the
form of the second square.
Suppose the second square is the
inverse of the first. Then k -I 011"~Ii h = f which
and
k~ll .011h -i = f'
and
k ~ l l h - o l I = f'h;
(6.19)
implies that
Oll-k~ll h = kf i.e. that Conversely,
k~llh = 01 i
-I
. Hence
if 011 -i exists,
i) ~
then
ii). Similarly,
~Ii = k -I 01 i -lh-I
(6.19), so the c o m p o s e d squares have the forms
?
7
7
151
7
i)--> iii). satisfies
1,6
Since transposes
are unique,
identity squares.
Hence
i)
Ad-Fun
these must be the corresponding
ii) and,
1,6.10 Proposition. X ~
in fact, extend to triple
Proof:
The first double
superscript while double
iii)
functor takes an object
~u)
(f----4u) to u,
In the second,
it refers to the 2-category (f
functors
o
on the left refers to the vertical
functor takes
double
functors.
square to (k,~22,h).
on the right
imply i).
o
~[OPFun opt]
which,
and an adjoint
similarly,
There are "forgetful" ) (Fun
ii) °P(Ad-Fun
17
the "op"
composition, structure.
to f and an adjoint
The
square to
(h,~fl,k). 1,6.il.
Examples.
I. The bijection
(6.8)
in 1,6.5 corresponds
to adjoint
squares of the form k
C
~B
ci[cr I flu C
while
the bijection
h
(6.9)
~A
is given by taking
the right
side to
consist of identities.
2. In studying composition
fibrations
later, we will need horizontal
to show that the composition
152
of cartesian morphisms
I ,6
is c a r t e s i a n . i.e.,
There
fibrations
is no o t h e r
in C a t °p.
3. A s q u a r e
of the
i8
proof
This was
for c o f i b r a t i o n s
overlooked
in
in Cat;
[FCC].
form
.~ B t
B
U t
g'
A
uf
then a 2-cel!
e: v'u'
>uv
has
transposes
(evf') • (fef') "(fv'~]') : fv'
.~vf' (6.20)
e =
(~f,g,) - (g~,)
while
a 2-cell
• (gf~,) : g f
~: u v -
~f,g,
~v'u'
has
transposes
=
(~'u'g) - (g'~g) - (g'u~) : g t u
)u'g
=
(e'gf)" (f'~f). (f'g'~) : f'g' -
~gf
(6.21)
It c a n b e c h e c k e d if g' f' ~
= v' gf
i) a n d
= A a n d u'
which ii)
that
of
if
~ = e
= uv then
coincides
with
then there
the
1,6.3.
153
~ = e
. For
is a t r a n s p o s e
isomorphism
instance,
isomorphism
given by combining
1,6
4. M a n y of the more or more generally,
interesting
Kan extensions,
1.6.i2 D e f i n i t i o n .
Let
s for X is a functor E
The w e a k l y dual n o t i o n
s: A
Es,x:
considered Note:
~B
Given
be a m o r p h i s m
limits,
in a
in A. The riqht Kan e x t e n s i o n
A(A,X) ~
is c a l l e d
}A(B,X)
such that
left Kan extension,
i.e.,
I E s'x
duals have no names
in Cat.
involve
~ ~(s ,X)
s,X
a(s,X) The two strong
examples
to which we now turn.
A, and let X be an object
2-category along
19
since
they have never been
(see 1,6.14(5).)
h: A
~X, k: B -->X,
this
says
there
is an iso-
morphism
A(B,X) [E s , X h , k ] and that these
isomorphisms
h and k by 2-cells. all,
c A(A,X) ( h , k s ) are natural w i t h r e s p e c t
For a g i v e n
or m a y be defined
s and X, Es, x need not exist
only
for certain
values
• ¢ F u n ( A °p)
x A
denotes
full
determined
by objects
(s,X)
is chosen
for all such
follows
follows
are,
of course,
1,6.13
Theorem. E
which
valid
There
: ¢ --9
vertical
E
s,X
if E
s,X
of h.
exists. cases
subcategory
We assume
defined.
is an o p e r a t i o n Ad-Fun (Cat)
pseudo-functor
composition.
154
(I,3.2)
it
of what
is only p a r t i a l l y
to h o r i z o n t a l
at
In what
(triple)
Many p a r t i c u l a r
is a 2-functor w i t h r e s p e c t
and a h o m o m o r p h i c
the
such that
(s,X).
to varying
composition
with respect
to
I ,6
Proof: Consider
20
a morphism m
A'
.j~-A f j/
s'
X e"
B v
s
"
n
~B
Y in • (i.e. , Es,x and induces a natural ~.:
Es, ,y) are defined.
e: sm
-~ns '
transformation
A(m,f) oA(s,X)
whose component
Then
at
>A(sV,y) oA(n,f)
k e A(B,X)
(~*)k = fke:
is
fksm ,
We define the value of E
~fkns'
on this morphism
to be the adjoint
square A(n,f)
&(B ,x~
> A(B' ,Y)
J
';f E
s,X
A(s ,X)
Es, ,y
A(A ,X)
A(s' ,Y)
A(A' ,Y)
A(m ,f)
Thus Em'n;~ s,s';f
: Es'
,yOA(m,f)
is the transpose natural If the adjunction
"
>&(n'f) °Es,x
transformation natural
es,X:
es,X ° A ( s ' X ) - > A ( B ' X )
qs,X:
A(A,X))
~..
transformations
&(s ' X)oE s,X
155
to
are denoted by
1,6
21
with components (eS ,X)k: Es ,x(ks) (Ns,X) h : h
>k > (E s ,x h) s
then its component at
h ~ A(A,X)
,
is the 2-cell given by the
composition Es, ,y (fhm)
(6.22)
E s, ~y[f(E s,X h)ns'i
)f(E s,X h)n (es, ,y) f(E s,xh) n
It is clear that a 3-cell in ~ given by taking compatible 2-cells for m,n,s and s' and an arbitrary 2-cell for f determines 3-cell in
Ad-Fun I A I
and that E- is a 2-functor with respect
to horizontal composition,
since the
~,'s compose properly.
With respect to vertical composition, we
only obtain a homo-
morphic pseudo-functor since there are only canonical isomorphisms Et,X ° Es,X
Ets,X
rather than identities. 1,6.14 Examples. In the following special cases any constituent of E- which is an identity is omitted from the notation. The purpose of these examples is to illustrate the contention that transposes are the general case of a "canonical
156
1,6
induced m o r p h i s m these canonical properties.
22
" and the point of 1,6.13
is that, therefore,
induced morphisms have all possible n a t u r a l i t y
We have not investigated
for bicategories,
the corresponding
situation
but there does not appear to be any obstacle,
other than finiteness
of available p u b l i c a t i o n
space,
to
doing so. I) . If
f
~ u, then for all X, Ef, x and E u'x
exist
and we may always assume they are chosen to be Ef,X = A(u,X) ,
E u,X = ~(f,X)
by 1,6.4.
2) Definition.
Es;f
f: X
>Y
preserves
E
S~-
if
d~f Ks,s;f: Es,y"~(A,f) - ~(B,f)o Es, X (6.23) E s,Y(f(-))
is an isomorphism. (Cf., Dubuc
=~ fE s,X (-)
The following
are
[ 9] and Gabrie!-Ulmer
a) If f and g preserve Es;qf =
E
S~-
immediate
from 1,6.13.
[17]-) , then so does gf and
Es;qn"] Es; f
b) If f preserves
Es,_ and Et, - , then it preserves
Ets _ and Ets;f = E t ; f ~
Es; f
c) If f has a right adjoint u, then f preserves for all s. This
follows by applying
diagram
157
1,6.11,3)
E
s~-
to the
1,6
TI . .
A(B,X)
Es ,X
.
A(B,f)
E s ,y
A(A, f) A(A ,u)
X(A,u) oE(s,Y) d) T h e o r e m u: B.
............
~(s ,Y)
--> A(A,Y)
= A(s,X)oA(B,u).
(Dubuc ~A
Ij
'~ A ( B , Y )
~(s ,X)
~(n,x) f°Ef,A(A)
Since
Ef,B(f)
we can and will
is d e t e r m i n e d
assume
that
o n l y up
(Ef,f) A is the
identity. Suppose Ef, X = A(u,X)
first that u: B---> A exists exists
Ef,B(f) Conversely, u = Ef,A(A). natural
suppose Then
in k and
for all X. In p a r t i c u l a r = A(u,X)(f) Ef,A(A )
there
are
exists
Ef,A(A)
and is p r e s e r v e d
isomorphisms
~ A(A,A) (A,kf)
A(B,B) (fu,fk)
~ A(A,B) (f,fkf)
A(B,B) (fu,~)
=" A(A,B) (f,~f)
158
f - - 4 u. Then = u
and
= fu = f0Ef,A(A) .
~, and a c o m m u t a t i v e
A(B,A) (u,k)
with
b y f. Let
for all k and ~,
diagram
1,6
24
Let k = u. Then lu corresponds to
qA: A
corresponds to
qf = fNA" Similarly,
qf: f
}fuf
and
= B, if corresponds to gB: fu 2-cell
~B.
SB' one gets the corresponding
}uf, while Ifu
If ~ is replaced by the commutative
fu
triangles fu f
~ ? I c
in A(B,B)(fu,-)
for
n f = / 7
and A(A,B)(f,-)
I
which shows that e and q satisfy
one of the adjunction equations.
For the other, observe that
the diagram A
q
~uf
uf .........quf
)ufuf
u~f ---->uf
commutes, the square by the functoriality of composition and the triangle by composing the previous triangle with u. Hence, for the indicated 2-cells in place of k, one has corresponding commutative diagrams
qu
~]uf
A(A,A) (A,-f).
159
uf f
1,6
Hence
the other
e) there
adjunction
If A has
is e x a c t l y
identity h: A
2-cell
equation
a strict
one
2-cell
of a unique
is satisfied.
terminal
object
i
from A to i w h i c h i-cell
TA: A
(i.e.,
for each A
is therefore ~i)
then,
the
for
)X, we define l~m A h = E T A , X h:
Preservation
of c o l i m i t s
of p r e s e r v a t i o n above.
that
i
(i.e.,
~X
cocontinuity)
, and hence has s,are defined a n a l o g u o u s l y
Limits
of E
It is only under
E
25
special
the p r e s e r v a t i o n
assumptions
of c o l i m i t s
is a special
the same p r o p e r t i e s TA,X in terms of E
in a r e p r e s e n t a b l e
implies
case as
2-category
the p r e s e r v a t i o n
of
S~--
f) If A' called
is a sub 2 - c a t e g o r y
A'-cocomplete
if
E
of
exists
A, an object
X e A
is
for all A ~ A'. A m o r p h i s m
TA,X f: X'
bY
is called
A'-cocontinuous
if it p r e s e r v e s
E TA,-
for all A e X'. Given h: X
>X
where
k: X-
)Y
with Y
k': X-
>Y, unique
A' = Cat,
other theory
A'-complete
g)
there
and a clue
these
exists
up to an isomorphism, of
lines,
with
to the d e v e l o p e m e n t
note
a diagram
160
that
that
k'h = k.
small Hom sets
the Y o n e d a
see G a b r i e l - U l m e r
any
an A t - c o c o n t i n u o u s
such
large c a t e g o r i e s
In this connection,
for a m o r p h i s m
such that given
then this c h a r a c t e r i z e s
examples along
X, one can ask
X is X'-cocomplete
If A is the 2 - c a t e g o r y and
an a r b i t r a r y
imbedding.
For
of a r e a s o n a b l e [17],
§15.
if A has products,
then
I ,6
26
Pr A
>A
A x C
s × C
B xC
>B Pr B
gives
rise
to a d i a g r a m
,[
E
pr~,X
~(B,X)
A(B × C,X) L
Es
x C,X
~%(pr B ,X)
~(s x C,X) E PrA,X
~(A ,X)
A(A × C , X ) ~ A(pr A ,X)
in w h i c h
~%(PrA,X) oA(s,X)
1,6.11,3) , there
is an
Es,X ° with so
all
possible
pr B = Tc ,
which
× C,X)oA(PrB,X)
EprA,X
EprB,X °Es x C,X
naturality
properties.
On the o t h e r
by
interchange hand,
Specializing
(6.24)
to B = i
s x C = pr C ~ g i v e s
EprA, X = limcO
is the u s u a l
and hence
isomorphism
S = T A, a n d
limAo
= A(s
of
Eprc, X
limits.
the d i a g r a m
161
(6.25)
I ,6
27
X(pr B ,X) .......)A(B . × C,X)
~(B,X) L
E
A(s,X)
s,X
Es x C,X A(PrA,X) pr A,X
A(A,X)~
~(s x C,X)
j~ A(A × C,X)
E in general
gives nothing
unless we assume
E s × c,xOA(PrA,X) (which,
in 1,6.11.3)
identity between isomorphism.
is the e corresponding
(6.26)
to 8 being the
the two composed representable
function
) is an
Its inverse then has as transpose E
Taking
>~(PrB,X) o Es, X
s,X
o E
PrA,X ~
B = i as before,
PrB,X ~E
oE
this specializes
(6.27)
s × C,X to
PrA,X l!m Ao E
which
......
>l!m Co E
is the usual non-isomorphic
(6.28)
Prc,X
interchange
of colimits with
limits. In Cat, to write
(6.26) holds,
A(A x C,X)
Kan extensions can, of course, 2-category,
using the cartesian
~ A(C,A(A,X))
to write
closed s t r u c t u r e
and the enriched nature of
Es × C,X z A(C,~s ,X )" This argument
be generalized
to a suitable monoidal
closed
but other than this, we do not know the range of
validity of assuming
that
(6.26)
h) One can, of course,
is an isomorphism.
equally well talk about
162
f reflecting
1,6
or creating
limits or colimits.
3) Definition. s: A
Em s,sm~X
:
Esm,X
is an isomorphism~
W@ shall use this in Chapter
m: A' .... ~A
)B if for all X, Esm,X
28
is final with respect exists
oA(m ,X)
~E
i.e., there
IV.
to
iff Es, x exists and
s,X
is a commutative
diagram of
isomorphisms A(B,X) (E s,x h,k)
-~ A(A,X) (h,ks)
II
I~ x(m'X)
A(B,X) (Esm,xhm,k)
for all h: A
>X, k: B-----#X.
a) In particular, to TA: A
~ A(A' ,X) (hm,ksm)
m is final
if it is final with respect
)I; i.e., there is an isomorphism limA,[ (-)m] -~ l!m A.
Equivalently,
for all x : I ~,,
A(A,X) (h~XTA) b) Given n: A"
}X, there is an isomorphism
-~ X(A' ,X) (hm,XTA,) )A',
if n is final with respect to sm
and m is final with respect to s, then
mn
is final with respect
to s. c) If m has a left adjoint,
then m is final with respect
to s for all s. d) As in 2) , this can be used to describe in a 2-category.
For
instance,
if A has products,
163
various notions then D e A
1,6
is called d i r e c t e d
if the diagonal D
4) Another p o s s i b i l i t y n: B
~B w
29
%D × D
is given by
is final.
s: A
>B
and
w h e r e one can require that En s,ns;X: E s , X
be an isomorphism.
~ A(n,X) o Ens,X
We do not k n o w what this means.
5) The preceeding c o n s i d e r a t i o n hold in any 2-category. In particular,
they hold in Cat °p. We are not aware that anyone
has studied w h a t we choose to call dual Kan extensions; Kan extensions
in Cat °p. For instance,
situation. An example S: A
5~'
ES,I
(resp., right)
2)d) holds
is given by taking
(resp., ~S,~)
adjoint
(since
exists
X = ~.
i.e.
in this Given
if and only if S has a left
~ ! = A, etc.). We have not been
able to discover examples that are not v a r i a t i o n s of this one.
6) Kan e x t e n s i o n s provide a typical example of adjoint quasi-natural
t r a n s f o r m a t i o n s as in 1,6.2
(3), p r o v i d i n g we
r e s t r i c t a t t e n t i o n to the locally full s u b c a t e g o r y d e t e r m i n e d by m o r p h i s m s 1,6.13)
6' of
(as in the b e g i n n i n g of the proof of
in w h i c h ~ is an identity 2-cell;
i.e., sm = ns'.
Taking the b o t t o m row and top row r e s p e c t i v e l y of the value of E-
on such a square provides two 2-functors F,G:
i.e., F(m,n;f)
~t
= A(m,f)
>Cat; and G(m,n;f)
=
~(n,f) The sides can
be viewed as the c o m p o n e n t s of quasi -natural t r a n s f o r m a t i o n s b e t w e e n F and G
(i.e., m o r p h i s m s
164
in
°PFun(°P~',°Pcat)) , where
1,6
~s,X = Es,x: ~m,n~f
30
>A(B,X) : F(s,X)
A(A,X)
......
>G(s,X)
= E m,n s,st~f
~s,X = X(s,X) : X(B,X)
~X(A,X) : G ( s , X )
bF(s,X)
~m,n; f= id The adjunction
natural
transformations
provide
the modifications
follows
from
~11
(6.14)
between
Es, X and
e and q. The first equation
and the second
= id.
165
from
(6.13)~
A(s,X)
in (6.3)
in both cases
1,7
1
1,7 Quasi-adjointness. Basically, F: ~ m >
B
mations
~: F U - - >
equations.
and
a quasi-adJunction
U: B m > B
However,
~
and
between 2-functors
is a pair of quasi-natural q: ~ - - >
UF
satisfying the usual
it turns out that this direct generalization
the usual notion is both overly and insufficiently (but by no means all) useful formulations The analogues of 1,6.3 and 1,6.4 discussed
in 1,7.3 and 1,7.4.
to be "functorial"
characterization
are described
which is of particular
suggests another kind of quasi-adJointness,
described
in 1,7.7.
consists of examples,
Its relations
In 1,7.8,
universal mapping properties
Some
in 1,7.1. are
seems to indicate that the
squares is not appropriate
which is defined in 1,7.6.
of
The general failure of quasi-adJoints
here.
in terms of 2-comma categories
of quasi-adJointness
general.
(which fail to hold in general)
in the first variable
technique of adJoint
transfor-
a
is given for the type
interest here.
This
called transendental, to the previous notions are
the connections
are discussed.
of these notions with
The rest of the chapter
as follows:
1,7.9
Some general principles.
1,7.10
Some finite quasi-limits.
1,7.11
Quasi-colimits
1,7.12
Quasi-limits
1,7.13
Quasi-fibrations.
1,7.14
Quasi-Kan extensions
1,7.15
The categorical
1,7.16
The global quasi-Yoneda
1,7.17
Globalized
in Cat.
in Cat.
in Cat.
comprehension
scheme.
lemma.
adJunction morphisms.
166
In 1,7.5,
1.7
To begin, let between 2-categories,
F: ~ - - >
let
natural transformations
B
e: F U - - >
2
and B
U: B - - > ~ and
(i,2.4) and let
s
~: ~ - - > and
t
be 2-functors UF
be quasi
be modifications
(1,2.4, MQN) as indicated:
FUF F
UFU >F
F
U
U
'>U
One can form the composed modifications
~1
> UF
r~
(7.1) IIF
and
FU
FU
k
sU •
FUe
.I
(7.~,) FU
~ B E
Here, for instance,
q~
has components
167
q(~A)"
1,7
3
1,7.1 Definition. i)
The four-tuple
(e,~;s,t)
is called a weak quasi-
adJunction ii)
If
s
and
t
are isomorphic modifications
then it is
If
s
and
~
are iso-quasi-natural
(I,4.2~).
s
and
t
are identities;
if
called i-weak. iii)
then it
is called i-quasi. iv) and
Us
If
• ~U = U, v)
called and
then
(~,~)
(abbreviated
(7.1') are identities.
eF • F~ = F
is called a quasi-adJunction.
In any of the p r e c e e d i n g
strict
i.e.,
s)
situations,
the adjunction
if the composed m o d i f i c a t i o n s
For quasi-adJunctions,
(u~)(~n)
(7.2)
~ (F~U) = 1
We can thus speak of x-weak y-quasi-adJoints
x = -, +,
in (7.1)
these reduce to
= 1
where
i, s, si
y = n, i, s, si.
Here
-, +
adjective
denote the absence weak and
n
and presence
denotes no modifier
appears as a modifier only once,
respectively for quasi.
of the
Since
we have the following twelve
possibilities
168
is
s
1,7
-
+
i
•
•
@#
4
si
s
n
(7.}) i s
si
The
*,s
indicate
principle
the two cases which
ones; namely,
1,7.2 Definition.
that
are called
transformations
F~
and
i-quasi-adjoints
A pair of morphisms
between bicategories natural
i-weak
seem at the moment
Ue
(I,3.3)
are defined
As with 2-functors,
e: FU --> B (I.4.20)
there are various
and s-quasi-adJoints.
F: B --> B'
quasi-ad~oint
and
to be the
and
U: B' --> B
if there are quasiand
~: ~ --> UF
such
eF • F~ = F, Ue • ~U = U.
other possibilities,
but we do
not treat them here. For the analogue
of 1,6°3, F
,4 .~.. u
)
consider
2-functors
F'
8_
~ U'
with weak quasi-adJunctions
(E,~;s,t):
(e',~';s',t'):
1,7.3.
I U
F' --I U'
PropositigD i) a)
and
F
$: U - - >
There exist quasi-natural
U.
169
transformations
~: U w >
1,7
b)
If
t
and
exist modifications c)
~
5
are isomorphic modifications, then there
u: U - - >
~--~ and
~: U - - >
~
If, in addition, the composed modifications in (7.4)
and (7.4') below are isomorphisms, In particular, d)
then
u
and
~
are isomorphisms
this holds for i-weak i-quasi-adJoints. If
U
and
U
are i-quasi-adJoints and the compositions
in (7.4) and (7.4') below are identities, then
U
and
U
are quasi-
isomorphic. ii) a)
F'F
b)
(s,~;s,t)
If
i-quasi-adJunctions, c)
is weak quasi-adJoint to
then
and F'F
(e',~';s',t')
F,F
are both i-weak
is i-weak i-quasi-adJoint to
If both are quasi-adJoints and if for all
(Uh')~U, c = id
then
UU'
and iii)
,
F
C e ~, A e ~,
(F'e)h,F A = id
UU,
are quasi-adJoint.
If
F: 2-Cat o --> 2-Cat o
is a functor which is
enriched with respect to the closed structure given by then
UU'.
Fun(-,-),
preserves x-weak y-quasi-adJunctions.
Proof:
i)
U.
/t
I UFU
\
Consider the diagrams
~U
, UFU
U~
~ ~F"U"E~
,.-
/
~";' '~U
,,
,,,
170
UFU
us
*U
UFU
:,U
(7.4)
i,7
and (7.4') in which the roles of
=
(u~)(~):
6
U
and
~
u
U-->
are interchanged.
Define
.
The results then are immediate. ii)
UU'
.................
These follow directly from the diagrams
I]UU'
> UFUU'
t U / l ~
U~]'FUU'
) UU' F' FUU'
(7.5)
U'q' U"
UU'
>UU'F'U' Ut'
UU' e'
and
F'F
F'F~]
F' FU~]'F
~ F'FUF
~ F'FUU'F'F
(F' e)~3, 1 1
~
(z.5,) F'~F
~ F'U'F'F "~F'T]'F ~I ~ s ' y !. ~'F'F
I
iii)
Remark:
' eU'F'F
/
FTF
This is immediate.
Parts i) and ii) clearly admit many special cases other than
those specifically mentioned.
171
1,7
1,7.4. Proposition. y-quasl adJoint. i)
Let
F: ~ - - >
7
B
and
U: B - - > ~
be x-weak
Then
The pair of 2-functors Fun(/~l,F) Fun(/~/~) ~
) Fun(/~,B)
is x-weak y-quasi adJoint for every ii)
~.
The pair of 2-functors
°PFun (U,/~) °PFun(S '/n)~OPFun(F,/~) is x-weak y-quasl adjoint for every
Proof:
OPFun (;4,/•)
~.
Part i) is immediate from 1.7.3,
enrichment of the covariant hom-functor. (E,~,s,t): F - - I and write
U
F* = Fun(F,~),
K~: K F U - - >
K
in
°PFun(B.~),
Fun(~,~)
F
to
~
~FU
-
K c °PFun(B0~) ~: K m >
> K' FU
> K'
is a modification
operates similarly,
with components
172
U
ops.: OPu.oPF . __> OPFun(B.~)
Fun(B,~)
K
opt.
Then
transformation whose value on
KFU
direction,
etc.
and on a quasi-natural transformation
is the modification
Thus in
To check part ii), let
be the x-weak y-quasi adJunction from
is the quasi-natural is
iii) plus the usual self-
s*
going in the proper is the modification in
K'
1,7
8
KFUF
KF
so in
°PFun(~,~),
........ ~ K F
KF
°Ps*
goes the other way.
One checks easily then
that
(°Ps*,°P~*,°Pt*,°Ps*):
I °PF*
°Pu*
is the desired x-weak y-quasi adjunction.
Remark:
The occurrance
of the weak dualization
effect that the covariant and contravarlant quasi-adjunctions squares,
cannot be combined
as is the case with ordinary
in part ii) has the
instances of induced
into the study of quasi-adJoint adjunctions
(cf.,
It would be nice to have a characterization types of quasi-adjolntness hom-functors.
However,
in terms of properties
it is easily checked that,
of i-weak i-quasl-adJunctlons, complicated
adjunctions theorem,
W
×
s
of various
of Cat-valued except in the case
studying hom-functors
leads to a more
situation than that of quasi-adJunctions.
One of the purposes of the introduction categories
1,6.6 ff).
of 2-comma
is that they enable one to reduce the study of quasito the study of ordinary adJunctions.
we have 2-functors and Cat-natural
In the following
transformations
over
(cf., !,5.5) S (7.6)
A×B
173
1,7
~: ST --> id,
~: id --> TS.
9
Also.
~F : (1.l,F) j# A - - >
[F,~]
(7.6,) 7U : (U'I'I)Ju: B --> [A,U] (Cf.,
(5.3) and (5.7).)
(Note that if
$
and
~
are assumed to be
either quasi-natural or natural then it follows that they are Catnatural.) It will always be assumed that morphism
(i.e., opcleavage preserving)
morphism (i.e., cleavage preserving)
S
and
is a right U.-homoT
is a left F.-homo-
so that
S = [n.U] o U. T = [F,s] ° F.
for unique quasi-natural transformations
~: ~ m >
UF
and
e: FU --> B
(See 1,5.10)
1,7.5 Theorem. i)
There is a biJection between four-tuples
above and weak quasi-adjunctions
~Ju
and
and
between
F
and
U.
ii)
i-weak adjunctions correspond to four-tuples with
~JF
isomorphisms.
iii) ~Ju
(e,~;s,t)
(S,T,@,~)
$JF
quasi-adjunctions
correspond to four tuples with
identities
174
as
1,7
iv) and
~
Strictness corresponds to a four-tuple in which
define an ordinary (Cat-enriched) adJunction
Remark:
S
......I. T,
Thus strict quasi-adJunctions are equivalent to "homomorphic"
adJoint functors are over
~ × B
Proof: i) and
i0
E.
S-- I T
where the adJunction morphisms
and satisfy
By assumption
~U
S
B) ~ [F,B].
and
~
determine each other as do ~
and
s,
let
Then
TS(h) = eB(FUh)(F~A): FA --> B.
A natural transformation
@
over
=
where
~h
h
to
SA = ~FA : F A - - >
Computing
~
~ × B
h-->
is a e-cell from
FA
>B h
in
[F,B]
Define
(EFA) • (F~A).
> FA
FA
has components
TS(h)
TS(h).
on the morphism
FA
and
= id = ~JF"
To determine the relation between
(h: FA m >
~
shows that
175
T
1,7
Ch
=
Ii
(gh(FBA))
(7.7)
(hSA)
'
One calculates directly that this formula describes a biJection between natural transformations ~: F m >
(EF) ii)
(F~).
9: id --> TS
Similarly,
By construction
~
and
E
and modifications determine each other.
(~JF)A = (l,SA,1),
immediate that this is an isomorphism in
[F,B]
and it is
if and only if
sA
is an isomorphic 2-cell. iii) iv)
A
This follows as in ii). Let
~A
(h: FA --> B) ¢ [F,B]
and consider the diagram
, UFA
nnA UFA
UFA
~ Uh
t e A
"~~
--~ UFA
The clockwise and counterclockwise outer compositions from UB
are
S(h)
= (UB)(Uh)(~A) = (Uh)(UFA)(~A)
while in the middle there is the composition
176
(7.8)
A
to
I ,7
12
STS(h) = (UeB)(UFUh)(UF~A)(~A)
.
Furthermore
S~h = [((Ueh)(UF~A))
• ((Uh)(USA))]~A
: S(h) --> STS(h)
while
~S h = (tB(Uh)~A)
Thus,
if the quasi-adJunction
modification, adjunction S - - I T, large
it follows
equation then
diagram
then since
(7.i) that
similarly
(7.1) follows in which
is strict,
from
follows
• ((UeB)(~Uh~A)).
t
is a
$S h • SSh = id.
from
(7,1').
from the particular
The other
Conversely,
case of the above
h = id: FA --> FA.
Part iv) of the preceeding
theorem
suggests
the following
definition.
1,7.6.
Definition.
functors
F: ~ n >
A transendental B
and
quasi-ad~uncti0n
U: B - - > ~
consists
between
2-
of a pair of 2-
functors S
) [A,U]
IF,B] < T over
~ ~ B
$: ST----> id
such that and
if
S-- I T
9: id --> TS
via Cat-natural over
177
~ × B.
transformations
I o7
13
The difference from the preceeding notions is that and
T
S
need not be homomorphisms of any kind and hence need not be
given by composition with appropriate quasi-natural transformations. However, as in 1.5(5.33),
S
and
T
always determine quasi-natural
transformations via
---
.....
= (~
--
JF
One can ask when this adJunction.
> [u.u]
- - . >
[FU,~])
> [F,F]
- - - >
[.,4,UF])
~
and
~
are part of a suitable quasi-
Some of the results of 1,7.5 hold in this situation.
1,7.7 Theorem. i) and
~
Functors
S
and
T
and natural transformations
(no conditions) determine a weak quasi-adJunction
(e,~;s,t). ii)
If
~Ju = id = ~JF
homomorphisms G e e proof) then ill) then
(E.~;s,t)
Remark:
If
(S,T;~,~)
and if
(e,~)
and
T
are partial
is a quasi-adjunction.
is a transendental quasi-adJunction
is a strict weak quasi-adJunctlon.
Conditions ii) and iii) together thus describe when
is a strict quasi-adJunction.
Proof. i)
S
To define
s
and
t,
let
178
(E,~)
1,7
T
r~A
I
=
\IIFA
(~A~I TS(1FA)
:' UFA/
1
14
~
FA
~
1
/~ A/
EFA#
J
FA
(7.9) =
S \FUB
Set
S(~B~ ST(IuB
n
~ B /
~B
sA = h A " ~IFA
and
~B /. UB
tB = ~ B
" ~B "
Then
(s.~;s,t)
is a
weak quasi-adJunction. ii)
Consider the functors
< j F , ~ >: B
•> IF,F] ~ IF,B]
~,ju
-> [~/,U] ~ [U,U]
>. ,W
The natural transformations of 1,5.9 and 1,5,10 which express the failure of
S
and
T
to be homomorphisms,
when restricted to the
images of these functors have values which are the above, ~A'S
Thus we call and
~B'S iii)
S
and
T
are identities.
and
Part ii) is now immediate.
IF,B]. F~]~ -
FA
~
l
~B'S
partial homomorphisms if all the
The situation here is a bit more complicated.
Consider the diagram in
FA-
~A'S
~A
) FUFA
~
,~
T(nA)
;, FA
1
179
1,7
Applying
S
15
gives the diagram
A
~A
UFA
/t A A
.qUFA ~A
~. UFA
/ ~
(7.10)
IS(SFAJ/ ~A I ~A
i 1
\
FA /
~,, ~ \ ~
U~A - -
i~.
~
/ ~
UTnA
UFUFA
qS UFA
TUF A
A careful examination of this yields the result. adJunction equations give
~SIFA
•
S~lFA
= id
Note that the
while naturality of
gives (~luFA~A) • S(XA) = ~SIF A. The bottom of the cube is U(SA) while the right side is tFA. Finally, the cube commutes since it is S
applied to a commutative cube.
1,7.8. l)
Let
and
U,
Universal mapping properties (S,T,~,~)
be a transendental quasi-adJunction between
Then, for each pair of objects
A s~
and
an ordinary adJunction SA,B ) [A,UB] .
[FA,B] ( TA,B
180
B s B,
F
one gets
1,7
Note that
[FA,B] = B(FA,B),
adJunctions
etc.,
16
so that this gives ordinary
between the hom-categories
These adJunctlons as follows:
(but not the hom-functors.)
can be described by universal mapping properties
given
(h: F A - - >
B) ~ [FA,B],
then
~h: h - - >
is universal not only in the usual sense in the category but in the broader
sense that given any morphlsm
Ff
FA ,,,
B
in
TS(h) [FA,B 1
IF.S]
~ FA,
)B'
g
then there is a unique
f
A .......
), A'
UB
> UB' Ug
such that cells.
T~OSSh
= ~.
A similar universal property holds for 2-
This characterizes
and the values of property,
S
S
in the usual
sense that, given
on objects satisfying this universal mapping
then there Is a unique way to extend
such that
(S,T;~,~) 2)
is a transendental
Suppose now that
sense of 1,7.5~
familiar
properties of
and
e
and
~: ~ - - - ~ UF
to a 2-functor
quasi-adjunctlon. T
are homomorphic
~.
Let
and
~: F U - - >
situation above translates q.
a strict quasi-adJunction, satisfied by
S
S
so that they are given by composition
natural transformations The simple,
T
in the
with quasiB
respectively.
into quite interesting
For simplicity we assume that
(~,q)
is
and describe the universal property
(h: A --> UB)
181
E [~,U].
Applying
ST
gives
1,7
~A
(A
UFh
.> UFA
~A
h
U~B
> UFUB
and the diagram on the left below,
A
17
> UFA
which we read as on the right
A
~A
> UFA
~ UB ~]UB
> UB) s [W,U]
gh' >lJ~ ~ UeB UB
where
h' = e B ( F h )
and
weak quasi-adjunction,
k h = (UeB)~h.
then one would take
X h = (tBh)
1,7.8.1 P,roposition.
• ((UEB)~h)
Given any diagram of the form
A
~A
> UFA
h
~
......
then there is a unique 2-cell
(7.12)
Ug
~: g - - >
7 = ~h " ((U~)~A)"
Proof:
Note that if this was only a
Consider the diagram
182
h'
such that
i ,7 DA
A
U~FA /
UFh ~ ~ X / _
This commutes since
~
18
> UFA
tUFUg
~ , #
I----~~Ug
is quasi natural.
ILUB
(7.13)
The adjunction equations
give
Ueg ~ ~Ug = i
while strictness gives
(UaFA) ~B A
l
.
Hence, defining
= Eg~FT:
gives ~ = :h To
g --> 6B(UFh ) = h'
((UT)~A)"
show that
satisfies this equation.
~
is unique, suppose Consider the diagram
183
v: g - - >
h'
also
1,7
19 Fh FA
•> FUB
]1
FUFA
FU~ B
The prism in back commutes by hypothesis naturality of strictness
~.
and the one in front by quas~
Hence the other adJunction
equation give
v = Sg ~
equation
and the other
Note that the same result
F 7.
holds in the weak case; here one takes
T = [eg [] FTS
also that the property
is dual in the sense of
satisfied by
E
(gsA).
Note
reversing both 1-cells and 2-cells. 3)
Under appropriate
hypotheses,
property gives rise to a quasi-adJunction, the thing constructed pose
U: B --> ~
h': FA m >
UFA B
except that,
is a pseudo-functor
denoted by
FA,
such that given any and a 2-cell
A
~h:
BA
in
rather than a functor.
B
h: A - - >
(Uh')~A m >
> UFA
184
mapping
in general.
is a 2-functor and suppose that for each
there is an object, ~A: A m >
such a universal
A E
and a morphism UB h,
there is an as illustrated,
Sup-
~,7
2O
satisfying the universal property that given any other and
7: ( U g ) ~ A - - >
h,
there is a unique 2-cell
7 = k h " ((U~)~A).
Define
to
UB.
h = 1UB: UB m >
1,7.8.2 Proposition: (see below)
then
If for all
F
s
h'
t
F
so that
and
The hypothesis
(7.15)).
If
h'
h,
B
with
corresponding
h: A --> UB, h' = eB(Fh ) (I,3.2),
~
and
(I.3.3) and there exist
(e.~;s,t)
is a strict weak quasi-
U.
means that the correspondence
is given by composition
after
as the
and all
transformations
and
adJunction between Proof:
A
B
T: g - - >
extends to a pseudo-functor
extend to quasi-natural modifications
eB: FUB m >
g: F A - - >
with
e
in so far as is possible
m: A --> A,, define HA
A
between
Fm
and
~m
h
and
(see
by the diagram
-> UFA
A'
> UFA' HA,
i.e.,
Fm = (HA,m) '
and
~m = k~Am"
If
then by the universal mapping property, determine a functor (Fn)(Fm)
a unique 2-cell ~(A,A,) and
F~: F m - - >
--> B(FA,FA,).
~n ~ ~m
determine ~m.n:
Similarly, determine
the identity map
~: m - - > m
Fm,.
and
m'
is a 2-cell,
(~A '~) " qm
By uniqueness
Now suppose
this gives
n: A'--> A,,.
a unique 2-cell
F(n)F(m) --> F(nm). IFA
and the identity 2-cell
HA
a unique 2-cell ~A: IFA --> F(IA)
One verifies that with this structure. functor and
~: ~ - - >
UF
F: ~ --> B
is a quasi-natural
185
is a pseudo
transformation.
Then
1,7
Define
e
on objects and
UB
To define k: B - - >
~
~U B
21
t
by the diagram
>UKOB
on morphisms,
consider the diagram for
B'
~UB
UB
) UFUB
(7.~5)
~
B)~-I -'-.,u~ k /
The precise hypothesis of this proposition is that in such a diagram not only is ksB
and
kt B
which makes
EB,(FUk ) = (Uk)'
but also
determine a unique 2-cell E
ZUk = tB' ~ ~Uk"
Then
ek: kE B --> SB,(FUk ),
a quasl-natural transformation.
Finally,
consider the diagram
A
~]A
~ ~A
~A ~A n ~
u9" UIFA
UFA
/"
> UFU~ F
1~ U F A
186
(7.16)
1,7
Again,
(SFA)(FqA) = (qA)'
determine a unique 2-cell
so
22
IFA
and the identity 2-cell
SA: IFA --> (SFA)(FqA)
qA
which satisfies
half of strictness automatically and can be shown to be a modification.
The other half of strictness follows by putting
together the diagrams defining the definition of
Ft B.
SaB
and
sUB
Note that in this case,
modified so that the composed 2-cell goes from SBF(1uB )
and taking account of
and is to equal
~
(7.1') is to be SB(IFL~)
to
~UB"
Examples. 1,7.9. Some genera ! principles.
There are many special kinds of
quasi-adJoints and many special situations arise,
Some of this
bewildering variety is accounted for by the existence of the 2-
Fun(B,B~'~') described
categories of the form
in 1.2.~.
There
are essentially three possibilities which must be considered. i) mations
One or both of the adjunctlon quasi-natural transfor-
s: F U - - >
B
and
q: ~ - - >
UF
may belong to a 2-category
of this form. ii) the form
In situations where
Fun(~,~)
~
or
B
or both should be of
(e.g., Kan extensions or limits), the appropriate
2-categories may actually be of the form
Fun(~o~;~,~').
iii) Cases i) and ii) can be combined so that there are adjunctions of type i) between categories of type ii). In particular,
we shall adopt the following terminology for
limits.
187
1,7
23
1,7.9.1 Definition. i) A: ~ m >
Quasi-adjoints to the constant imbedding
Fun(~),
where
~
is a 2-category and
~
category are called quasi-limits and quasi-colimits;
is a small 2written
Adjectives from 1,7.1 will be added as appropriate. ii) Cat-adjoints
(type i), via 1,2.4i)) to a constant
imbedding as above are called Cartesian quasi-limits and Cartesian quasi-eolimits,
written
Cart q-l_lim~ Cat
~A - - I Cat
Cart q - l ~
If there is only an ordinary adjunction at the level of the underlying categories, we write
Cart qo-l_~im~ and
Cart qo-l~_m~.
(The
reason for the word Cartesian is that for Cat-adjoints the functors S
and
T
in 1,7.5 are 2-sided homomorphisms,
i.e., cleavage and
op-cleavage preserving.)
iii)
Still more specially, Cat-adjoints to constant
imbeddings
,4 ,4 (see 1,2.4) are written Cart q - l_~m~_is° ~ , limits.
> Fun(~.%; W,Ao) > Fu~(~,~-,,W, iso W) Cart q - l ~ _ i respectively;
d ~;
and
and similarly for inverse
Note that the ordinary Cat-enriched colimit is the same as
Cart q - l_~Im~_id~o,
while
Cart q - l ~ _ i s o
to
188
~o
is the Cat-adjoint
1,7
A-->
24
ISO - Fun(gj4)
(See 1,4.24.) As with ordinary limits and collmits, cartesian quasl-ones can be defined in a more global fashion as Cat-adJolnts. the colimit case.
(Cf. 1,1o13).
N: X - - >
(the taking
s
Let
S[2-Ca~,X]®
X
to
r~ : 1 --> X,
be a 2-category and let
(See 1,4.25)
means small 2-categories over XEX
We treat
X)
f: X - - >
be the Y
"name" 2-functor
to the I -cell
m
I --.>
I
X
and a 2-cell
~: f - - >
g
to the 2-cell
1
ff~>! ry~
\rg 7 X in s[ 2 - C a ~ ( ] ® .
1.7.9.2 Theorem.
Let
X
have small cartesian quasl-collmits.
Cart q - li T : s [ 2 - C a t ® , X ~
> X
is an enriched functor which is the left 2-Cat®-adjoint to
189
N.
Then
1,7
Proof:
25
The main thing is to show that
Its value on an object Q(F) = Cart q - ~ m ~
F: ~ - - >
F.
X
Cart q - lim), is defined here.
is, of course,
If
A
M
> A'
X
is a morphism
(where
m
is quasi-natural),
then the diagram
~A ) A~ Cart q - l_~im~ F
F
I
I
i r
~F ,M
> (Sq'
F'M
C a r t q - ~ m / t , F' ) o M tl
A~ Cart q - l i m ~ , F ' )
shows that there is a (unique)
Q(M,m):
Similarly,
induced map
Cart q - li>n~ F - - >
if
M
F
X
190
Cart q - l_~im~, F'
1,7
is a 2-cell,
26
then the m o d i f i c a t i o n
m
F -
~F'M
tM1
induces a 2-cell
Q(n,e):Q(M,m) --> Q(M'~')
~Ca~-adjointness satisfied by
is immediate,
Cart q - lim F >
since the universal is easily translated
isomorphism between the illustrated
>
mapping property into a functorial
categories
1
X
The property construction complicated examples,
yields an enriched calculation
First,
s[2-Cat@,X]®
which is not immediate
.
functor.
is that the above
The proof is a very
and the reader may want to skip ahead to the
we must describe
the composition
in
The composition of a pair of 2-cells
191
M
27
1,7
(7.17) X
(because we are dealing with a 2-Cat@-category) is a diagram KM
?(
The verification that this is a 3-cell (i.e., that equation 2.19 is satisfied) follows from the diagram
192
28
1,7 m
kM
> F'M
F
> F"KM
kM'
~
~
F"K'M
kM~
F" K' n
k,
K which commutes since
~
from a 2-Cat®-category discrete"
'
(7.19)
M'
is a modification.
Now an enriched functor
to a 2-category regarded as a "locally,
locally
2-Cat-category and then as a 2-Cat®-category must turn (7.18)
into a commutative
square (since there are no 3-cells).
Q(-) = Cart q-llm
to (7.17) gives
,~
Q(M,m) Q(n,e)
/
Thus, applying
Q(K,k) ~
Q(.g,q~)
Q(M' ,m' )
"
(7.20)
Q(K' ,k' )
and one must show that the diagram
~
CM ,Q(,
(~M,)~, )
_ ~tQ(~M, (~M)m)
193
(7.21)
1,7
commutes.
29
The proof of the commutativity of (7.21) when
can be done directly using the construction of
Q
X = Cat
given in 1,7.11.
In general) it is sufficient to show that (7.21) is the composition (in
X)
of (7.20), since that composition commutes.
shows that
Q
(This also
is enriched, rather than possibly quasi-enriched.)
The proof depends on the universal mapping properties satisfied by the various constitutents of (7.21).
We illustrate the step showing
that
Q(~,(~)m)
;
o Q(M,m)
.> AA% ~ -
mI
~6~m
F,M _
~v
= Q(~,k)
t
~
Q
(
K
M
'
(kM)m)
> F'M'
I /I
F" KM
-> F" KM'
195
1,7
31
It is sufficient to show that the desired equation holds after composition with
~F"
It follows immediately from this big diagram
(which the reader must imagine or draw for himself) that
i)
[A~,Q(K,k)M'] o [A~Q(n,e)]~F : [A~,Q(K,k)M,][(qF,)n;:~e] = [ ~
o (~Qk)M
][(nF,)n~e ]
while ii)
~4Q(Kn,kn~0)~ F = (nF,,)Kn~kn~e
But now for any
f: A - - >
B
.
(in particular, for any
the diagram
F'A
F'B
kB
> ~ , QF'
F" KA
~ , Qk
(~F" F" KB
'> ~4, QF" K
commutes, and hence
[(~Qk)M'](~F,) n = (~F,,K)n ~ kn
Similarly,
( ~ )
((~F"K)n) = (~F,,)Kn
These two equations show that i) equals ii).
198
nA: MA --> MA,),
1,7
32
An analogous calculation based on the diagram
F
i
F'M
\k,.
F'~M )F"K'M
~'KM
shows that
AAQ(~M,(eM)m ) = [A/~,Q(~,~)M] o [A/~Q(M,m)]
which completes the proof.
1,7.10 Some finite quasi-limlts. l) Comma categories are characterized by a diagram
(F1,F2)
> A 2 e
(7.23)
AI
>B_ F1
where
e
is a natural transformation,
property as that of 1.5.2. follows:
let
P
satisfying the same universal
This is a Cartesian quasi-limit as
be the category illustrated by
m
1
and
P_~
the subcategory
Jl
J2 > 0 < ~ 2
0 <m2.
A quasi-natural transformation
from a constant functor to an arbitrary 2-functor from
197
~
to
1.7
33
looks like
X
This belongs identity
to
2-cell,
Fun(P,P_~; ~ , A o ) in which case
go = f292
g2
X
AI
We conclude in Cat.
72
is an
and the diagram reduces
.> A 2
fl
that a comma category
This
arbitrary
if and only if
'> A°
is the same as a Cart q-l_~mp_idP~
serves as a definition
of a comma object
in an
2-category. 2)
Subequalizers
are discussed
by Lambek in [27].
Here,
one is given a pair of functors
A
F
~B
G and one looks for a best possible natural
transformation
e: GM ~ >
M: M FM.
--> If
together E
denotes
with a
the category
m
o --X--~ i
and
E_~
the subcategory
that a subequalizer
0
S
> i.
then one concludes
is the same as a Cart q-limE_i d E_°
(Cf., 1,7.12.3).
198
as above
in Cat.
to
1,7
3)
34
Products and coproducts, since they are taken over
discrete categories, do not involve 2-cells, so Cartesian quasiproducts and coproducts coincide with ordinary ones; i.e., if discrete, then
Fun(~)
=~g.
~
is
On the other hand, strict (weak)
quasi-products and coproducts correspond to interesting universal mapping properties. coproducts, given
(Cf. 1,7.8 (3).) A1,A 2 e ~,
property that there are maps
then
A1 ~ A2
ij: Aj ~ >
that given maps
hi: Aj --> X,
and 2-cells
as illustrated
kj
For instance, in the case of
A 1 ~ A2, J = 1,2,
then there is a map
iI A1
should satisfy the such
h: A 1 ~ A 2 --> X
i2 > A 1 ~ A2
(
A2
(7.24) hi
~hi
I~ ~ X 2
h2
X
g: A 1 ~ A2 ~ >
with the property that given any other ~j
as indicated, then there is a unique 2-cell
~j
(~lj) ~j =
X
and 2-cells
~: g - - > h
with
.
Quasi-products satisfy the dual situation in which the 1-cells are reversed.
(One can also reverse 2-cells.) For instance, in the 2-category whose objects are sets,
whose 1-cells are relations, and whose 2-cells are inclusions of relations, the ordinary product of sets wlth its projections becomes such a quasi-product.
More generally, in Puppe, Korrespondenzen in
199
1,7
35
abelschen Kategorien, Math. Ann. 148 (1962), p. 1-30, a product is, by definition, 4)
such a quasi-product. If
~
is the
pullback diagram of part l)
can consider quasi-adjoints with domain either
~
or
then one Fun(~.~)
giving diagrams like
p B
is a 2-cell in
~'((F~)a)
of 1-cells
whose objects
are pairs
whose morphsism are pairs
and whose 2-cells are of the form
Composition
and hence
here.
Recall that
(A,a)
F.
isomorphism between
cocartesian morphisms,
However,
to
[1.F] = q - l_l_Im~F. The proof
there can be extended to show that the relevant 2-comma categories preserves
category)
~
in ~:
= ~.
and
~: F(f) a - - > b
(f,~) --> (f',~')
such that
is given by the formula
201
~
1,7
37
(g.@)(f,~) = (gf.@((Fg)~)).
This 2-category is the 2-opfibration fibres associated to [1,F]
F.
LVo[1,F ]
by making all the 2-cells
LVo[1,F ]
(1,2.9) with locally-discrete is the category constructed from
?
identities.
denote the canonical projection.
opfibration associated to JA: FA --> [1,F] JA(e) = (1A,$).
F,
(7.27)
Let
Since
QF: [1,F] m >
[1,F]
is the
there are inclusion 2-functors
of the fibres given by
JA(a) = (A,a)
and
Similarly there are natural transformations
ef: JA --> JB (Ff)
corresponding to
f: A --> B, whose components are
the cocartesian morphisms
(el) a = (f, iF(f )a). Define
~
to be the transformation
constant 2-functor
[1,F]
This is not quasi-natural
from the 2-functor
whose components are the since, given a 2-cell Ff
[S.,F] does not commute.
One has
[(~B ;~) " ~f]a = (f'(F~)a) while
202
~,
F
JA'S
to the and
the diagram
ef's.
1,7
38
(~f')a = (f''IF(f')a)
However
~
"
is a 2-cell from the first to the second since
v: f --> f'
and
1F(f,)a(Fx)
(Undoubtedly,
this is a 3-dimensional
but we are not concerned discrete,
a = (FT)a
quasi-natural
with such things here.)
then this is quasi-natural;
otherwise,
If
transformation, ~
is locally
define
nF = %~: F--> A(L~o[I'F]) Then
~F
natural
is quasi-natural. in
F,
It is easily checked that
so that one has a Cat-natural
~F
is Cat-
transformation
n: Fun(;4.Cat) --> A(L~o[I,-]).
It is sufficient property making Let
to show that this satisfies the universal L~o[1,- ]
X s Cat
the left Cat-adJoint
and let
~: F --> &X
to
mapping
A.
be a quasi-natural
m
transformation.
,B
If
f: A m >
B
in
~,
then one has a diagram
,F]
....
(7.29)
X
203
1,7
39
and we must show that there is a unique functor making all such diagrams commute. o JA = ~A
(i.e., on the fibre
Define FA,
~: LVo[l,F] m >
9: [I,F] -->
~ = ~A )
X
by setting
and on cocartesian
morphisms,
~(f'lF(f)a)
This determines
~
(7.3o)
: (~f)a
uniquely since any morphism in
[I,F]
has a
canonical decomposition
(7.}1)
(f,$) = (iB,$)(f,iF(f)a).
Since
$
is quasi-natural,
does commute and hence obviously unique. IK1,F]I
the diagram corresponding to (7.28) for ~
determines a functor
(Note that since
~
can at best be a transendental
~
which is
above is not quasi-natural, quasi-limit.
This has not
been investigated.)
1,7.11.2
Corollary.
The canonical projection
P: KCat, B ] --> Cat
creates cartesian quasi-colimits.
Proof:
Here
[Cat, ~_~ ]
~
: 1 --> Cat
is the name of an object
has 2-cells that look like
B
204
B
s Cat,
1,7
and
P
40
reads the top line. An object
same as a quasi-natural
transformation
F:~
,> Cat. Hence there is an induced
(7.29)
(with ~
replaced by
Fun(~,[Cat, rB~ ]) general,
if
from
f: X
~
B ) to
~ ~ F u n ~ , [ C a t , r B ~ ]) is the ~: F
> A B, where
~: L~o[I,F ]
,>B,
can be read as a morphism
and in
A V- It can be easily seen that,
)B, then a morphism
from
~
to Af
in
is des-
cribed by diagrams FA F f /
~ ' A x
B
where
~': F --> Z~X
~: (Af)~' -->
$
J
is a quasi-natural
is a modification.
~' : L~o[I,F ] --> X
and
~
~'
transformation
and
induces a unique functor
a unique natural transformation
that
L~o[I,F]
~X
B
is the desired unique morphlsm
in
[Cat,
: Cart q-l_~im/~ ~F.
205
rB_~ S.
Hence
[
so
I -7
1,7.11.3
Corollary.
is cocomplete
Let
~: ~ - - >
(resp., complete).
lim L~o-~ (lim
(lim FA
~A ) =
41
[Cat,
r~ ]
as above, where
Then
~A ) :
lim
lim .---). L~o[I,F ]
(resp.,
lim
~)
Remark:
This says that a colimit of colimits of diagrams can be
computed as the single colimit of the quasl-colimit
of the diagrams.
(Note that the maps between the colimits must be induced by maps between the diagrams.)
The same holds for limits of limits, except
that one still forms the quasi-colimit
Proof.
(See [CCS], §8, Example 7).
Fun~,[Cat,
of the diagrams.
Consider the diagram
Cart q-lim rB~ ]) ~ ~
> [Cat. FB~ ]
(7.33) W : Pun(A,B)
B
Here b E B B
N:
L~O~ B
B --> [Cat, rB~ ] to its name
%1: ~ _ _ >
is the "name" B
and a morphism
to the morphism
1
funator taking an object
1
;i
B
206
$: b --> c
in
1,7
in [Cat, N
].
rB7
As is well
has a left adJolnt
colimit in
B,
lim
--
adJoint to
li~ f.
~2
known,
B
is cocomplete if and only if
which assigns to By 1.7.4, F u n ~ ,
f:
li~
X --> )
B
its
is then left
X
Fun~,N).-- It is immediate that
so the diagram of left adjoints commutes, up to an isomorphism. second formula follows from the first by replacing
Remark:
B
by
The
BoP
There is a problem about size which we have ignored in the
above discussion. category
Cat s
such that if
To take care of it, assume
of small categories ~
instead of
contains a sub-~-
(e.g., take a two-stage universe)
is a small 2-category and
L~o[1,F ] ~ Cat s .
Cat
F: ~ - - >
Cat s ,
Then the corollary above should have
[Cat,~B ~ ],
then
[Cats. rB_~ ]
as well as the hypothesis that
~
is small.
From the construction in 1,7.11., two other kinds of Cartesian quasi-colimits let
~o'
can be determined.
be a subcategory of
~o"
ZF,/~o, = { ( f . l F ( f ) a ) i.e.,
Z F ~ o,
Let
F: ~ - - >
~ o'
Let
~ [1,F]
f E/~o'}
2F,~o,
be the image of
!
If
~o
= ~o'
we omit it from the notation.
1,7. Ii. 4 Corollary. i)
Cart
q - lim~-iS~o'~ v.
and
Let
is the class of all cocartesian morphisms of
over morphisms in L~o[I,F I.
Cat
F = L~°[l'F][~l~'~'o']
207
[I,F]
~Z F ~ o'
in
1,7
43
ii)
Cart q - l i m ~ _ i S O ~ o
ili)
Cart q - l l m ~ . i d ~o' F = L~°[I'F][[ZF:~o']]
iv)
Cart q - l_~im~_id~o F = L~o[I,F][[Z~I]]
F = LVo[I F][ZF I]
In particular, if
is locally discrete then
Remark:
Here
denotes the usual category of fractions and
[[Z-1]]
denotes the category in which the maps in
[Z -1]
lim F = [I,F][[ZFI]].
identities (i.e., coequalized with their domains.)
Z
are made
Case ii) is the
one considered by Giraud [19].
Proof.
These results are evident from the construction of
~
in
the proof of 1,7.11.1, by equation (7.30).
1,7.11.5 i)
As a simple example, consider the cocomma category
construction as described in
[FCC].
Given functors
i = i~2., then the cocomma category
Fi:
~o ->
~i'
is the co!imit (in Cat)
of the diagram
A A-°× ~o -o
......
A × ::
> A × 2 ~
(7.34)
o
It is dual to the comma category as in i,7.10.
(7.23), satisfying a
universal mapping property of the form
A
F2
>A
--o
2
(7.35) A1
>
Cat
such that
F(i) = Q', i = 1,2.
Q' is the object of Cat which looks like
[I,F]
F(0--> l) : ( r21:!
-->2)
F(0--> 2) : ( r f : !
-->S').
looks like
F(x)
[I,F][ZF I]
and
F(O) = ! •
lim F
F(O)
F(2)
looks like
= [I~F][[ZFI]]
looks like
/Y/Y 209
Q,
and
1,7
Adapting 1,7.ii.3 to this case, where [Cat,
rBf ]~,
such that
one sees that if
PH = F,
45
Fun(~,[Cat, rB~])
H: ~ >
[Cat,
rB~ ]
becomes is a functor
then the colimit of a diagram of type
li~ F
in
is given by computing successive pushouts as indicated
F(1)
F(2)
1,7.11.6 Proposition. quasi-colimits is Proof:
The closure of Sets C
Cat
under cartesian
Cat.
We give two proofs of this important fact.
The first shows
that finite cartesian quasi-colimits are sufficient if those of type ii) in 1,7.9 are allowed, while the second shows that "cartesian quasi-codense" l)
D
and a subcategory
m
such that, given any
B E Cat, there is a functor
FB: ~ - - >
with
Cat
is
in Cat.
We shall construct a finite category
Sets C
[~}
Cart q - ! i ~
D --o
-id DO F ~ =
is Just an initial part of the category I,i.4 and 1,1.6).
A_°p
of ordinals
(See
It is generated by objects and maps as illustrated eo do ~ e1 < dl ~ ~2 i < ....... 2 < 3 s~ t > s2 >
210
I ,7
46
satisfying the following equations (which are chosen to fit categories rather than simplicial objects):
dot = dlt = i doeo = doel = fo (7.36)
dleo = doe2 = fl dlel = dle2 = f2 els I = ees I = e o S e = els e = i eoS I = td o
ees e = td I
sit = Set = u
o
is the subcategory consisting of
[ do' eo' el' fo' t. Sl, se ,u]
Given
B
e Cat,
define
FB : ~
--> Sets
by
F B (n) = I_]~I n = 1,2,3, Fh(di) = l?il
, FB(t)=
IB~I
(7.37)
FB(eo) = IB~I, FB(e I) = IBVI, FB(e e) = IBJI FB(Sl) = iB[~°'--2}l, FB(S2) = IB[--2,~I]I
The notation is as in 1,1.6 and I,i.7.
The opfibration
[I,F] has
(three) discrete fibres consisting of the sets of objects, morphisms, and commutative triangles of instance,
cocartesian
B.
respectively,
There are, for
morphisms connecting each morphism of
B
its domain and codomain and each triangle with its three faces.
211
with If
1,7
A
a
47
)B
C
is a commutative
triangle
in
B_,
then in
[I,F][[ZFID_o ]]
there is
a diagram
a
>A
b
Hence the objects This forces
~, = ~
T,a,c,
transformation
~B
and
A
and hence
(Note that associativity from
.........~......C.......
F
takes FB
are identified,
~K
= ~
.
B
is given by
212
b
and
B.
Therefore
care of itself.) to
as are
The quasi-natural
which is the adJunction
morphism
1,7
2
IB
48
°1
~
.
i IB~Oa I
(7.39)
IB_I Here all components of
~B
on morphisms are identities except
(~B)dl , (~B)c2
end their composition,
I~I
f e IBm21,_ then
in
B.
If
T e I_B~31 as above, then
i
[(~B)dl] f _
[(~B)e2] T = a
denotes the inclusion of = f,
while,
if
and one has
[ (nB_~1 e2 IT -- c II [(nB_)dIIB~I]T " [(nB)e2lT = ba .
2)
To see that
Cat, recall that if
[!]C
~: ~ >
!
SetsC !
> Cat, then
e Cat, then the canonical projection course constant so observed in
Cat is "quasi-codense" ~i,~4] ~ A.
P: [i, rB_~] ~ >
~
(IP: [I, rB~] __> Cat) = I[i r~ ] .
[FCC], §6, [ l , r ~ ] ~ B
in If
is of But, as was
and hence
Cart q - llm[l r~ lIP = [l. rBJ] ~ B .
We retain the quotation marks and forgo a formal definition since there are several other possible meanings for "quasi-codense".
213
1,7
1,7.11.7.
Remark.
description
These results
of a 2-category
~
49
suggest that,
in the ultimate
which is "sufficiently
like Cat",
there should be some relation between
i)
cartesian quasi-colimits
il)
the construction
iii)
of "objects
of fractions"
in
the position of the subcategory
of "locally
objects"
properties
1,7.11.8 Co rollarT.
in
Let
category to a cocomplete
~
, or, possibly,
G: B g > category.
~
discrete of
1 s
be a funetor from a small Then there is a coequalizer
diagram
,I I G(~of)
> 1 I Q(B)
I_B~2[
.~ lira, , , G
IB__I
(and dually.)
Proof:
Let
> Sets be the functor constructed
FB:
in the first
w
proof of 1,7.11.6.
G
determines
//~/ilB_~oI
a diagram
\
f"~ (7.4o)
c
214
1,7
50
Apply 1,7.11.3 to this situation, where F(i) are discrete, li~
G
so the terms
is the colimit over
~ = G, etc.
~,i~)~ i
D
The categories
are coproducts and hence
of--the diagram
B
I
I G(3o~T) c
~ I
I G(8of)
(.....
~ I
(7.41)
I G(B)
It is easily shown that the induced maps are the usual ones and that the subcategory
251
is cofinal in
1,7.11.9 Kleisli cate60ries.
D,
which gives the result.
The previous examples have all involved
functors whose domains are locally discrete.
In the following example,
which is an adaptation of a result of R. Street. [39], the 2-cells play a crucial role.
As at the end of 1,4.23, Street considers
copseudo-functors from
1
to
A
as cotriples on objects in
A.
m
These are the same as 2-functors from with a single object) to
~.
~op (regarded as a 2-category
We shall show that for
~ = Cat, the
coKleisll category is the cartesian quasi-colimlt of such a functor. For our purposes the standard presentation of a cotriple is more useful. 1-cell
g
5: g ~ >
Let ~
be the 2-category with a single object
and all its powers g2
and
E: g ~ >
*
gn,
*,
and generating 2-cells
satisfying the usual cotriple equations
5g • 5 = g5 • 5 , g E-5 = ag • 5 = i,
A 2-functor a functor 5: G - - >
G: ~
~>
Cat
G = ~(g): A --> ~ G 2, E: G ~ >
objects the objects of
~
is determined by a category
A ~ Cat,
and natural transformations
as usual. ~,
a
The 2-category
as 1-cells from
215
A
to
[1.~] B,
has as
morphisms
1,7
51
GnA -->
in
Ao
B
and 2-cells generated by commutative
5A
GA
-> G2A
\/
eA
GA
B
L~o[1,G],
as a morphism in identified
with
every 1-cell from
~
from
h EA.
GA
to
Cart q
If
1,7.11.10.
F: A m >
B
in
~: 2 Cat.
G.
then one obtains
a triple in
~
and this
category for the triple in
~.
cofibrations. ....>.. Cat Since
Cart
This category
can be expressed
ordinary morphisms being
category of the cotriple
Associated
B
~ = L~o[1,~]
then yields the Kleisli
Let
to
so
-,lim>
~(*) = ACP,
construction
B,
A
It is easily check that composition becomes
the ordinary Kleisli composition,
is the coKleisli
-> A
\/
B
Hence in
triangles
be a functor whose value is a functor 2
q - lim¢
clearly looks like
[1,F](A,B)
is locally discrete,
= [1,F].
A IIB
= B(FA.B),
together with extra hom sets
[1,F](B,A)
216
= ~ .
z ,7
It can also be described
52
as the pushout in
Cat,
F
A
~B
A x 2 ,
Hence
[I,F] =
in the sense of
> [1.F]
is the universal
[FCC],
§5,
Thus this universal
only the right adJoint to the inclusion into categories under
A,
property of a cartesian
cofibration
cofibration
with
F,
is not
functor of split cofibrations
but it also satisfies
quasi-colimit,
associated
the left adjointness
i.e.,
Fj\ A
B
;
To get the dual construction,
one checks that
[I,F°P]°P T .
1,7.12.
Quasi-limits Let
category.
Let
in Cat.
F: ~ - - >
Cat be a 2-functor,
P: [I,F] --> ~
where
~
is a small 2-
be the canonical projection
define the category of sections of Cat)
217
[I,F]
and
to be the pullback
(in 2-
1,7
53 A
F[1,F]
> [1,F~'
....
(7.42) 1
>
It is easily checked that are 2-functors
S: ~ m ~
F[1,F] [1,F]
are natural transformations that if
A s~,
then
is locally discrete.
such that
PS = Id
~: S --> S' S(A) E F(A)
S(f) = (f,gf): (A.S(A))
and its morphisms
such that and if
Its objects
P~ = id.
f: A --~ B,
Note then
.....> (B,S(B))
where
(Ff) (S,(A)) Naturallty for ~A
~: S - - >
are maps in
F(A)
S,
s(B).
means (by (7.21)) that the components
and make the diagrams
of
(Ff)(S(A))
->
S(B)
~B
(Ff)(S' (a))
-> S,(B)
commute. If
~o'
is a subcategory of
Cart~o , r[1,F] for the full subcategory of
(resp., F[I~F]
218
~,
then we write
Clio, r[1,F])
determined by those sections
S
1,7
such that f e ~
of
is an isomorphism
.
Note that the morphisms
precisely
the cartesian morphisms
are the chosen Hence
cartesian
the first
the second consists
1,7.12.1
(resp., (f,o) in
morphisms
subcategory
above
of cleavage
54
the identity) with
[1,F]
~
for all
an isomorphism
while those with
in the given cleavage consists
preserving
of cartesian
of
are
c = id [1,F].
sections
while
ones.
Proposition. i) ii) iii)
In particular,
Cart q - ~ i 5 4 F = F[I,F] Cart q - l.!m~_iso~
,F : C a r t ~ , F[I,F] o o Cart q - l.~Im~_id~,F : Cart~, F[I,F] o o if
~
is locally
limF=
,(
discrete,
c%4
r[z,;]
then
.
o
Proof:
Define
components
EF: A F[I,F] --> F
[(~F)f]s
S
whose
are
(£F) A = evA:
Since
to be the transformation
= of
is a 2-functor,
is a quasi-natural
F[I,F]
--> F(A)
.
eF
is quasi-natural.
transformation.
Consider
219
Suppose
the diagram
~: AX ~ >
F
1.7
55
X
(7.43) FA
~JB
~
g
F
~ FB
If
X c X,
define
(~(X))(A)
=
(A,~A(X))
(7.44) (~(X))(f) = (f.(~f)X)
Then
~(X)
is a section.
~(t)A = ~A(t). eF~ = ~
and
Then ~
~(t)
If
t: X m >
y
.
in
X,
define
is a natural transformation.
is the unique functor with this property.
Clearly Hence
Cart q - ~im~ F : F[I,F]
The other cases follow immediately from the formulas given above.
1,7.12.2
Remark.
There is a certain analogy with the case of
ordinary limits and colimits which is worth observing and which probably generalizes. i) A --
Let
F: ~ - - >
with discrete fibres,
Sets. lim) F
Then
[1,F]
is an opfibration over
is constructed by making all
220
I °7
morphisms (1-cells) in
[1,F]
56
identities,
while
lim F
consists of
(
all sections of il) over
~
[1.F S. Let
F: ~ >
Cat.
Then
K1.F]
with locally discrete fibres.
constructed by making all 2-cells in Cart q - ~ F
is an opfibration
Cart q - l i m ~ [1,F]
F
is
identities,
consists of all sections of
while
K1,F].
Corollary 1,7.11.2 has an obvious analogue but, as far as we know, this does not lead to anything useful along the lines of I,7.11.3.
One of the examples in 1,7.11.5 dualizes,
but 1,7.11.6 does not.
In fact, if
obviously discrete,
Sets~
limits.
1,7.12.3
so
Cat
m>
F[1,F]
is
Finally, 1,7.11.9 dualizes nicely.
We can now calculate directly the Cartesian quasi-llmits
Cat
with
and
F(Ji) = F i,
S: p m >
[1,F]
such that
S(Jl) = (Jl,f), S(J2) = (J2.id). F2(A2) = A o,
Using the notation there, if
F(i) = ~i
consists of sections
A-o
Sets, then
is closed under Cartesian quasi-
giving comma categories in 1.7.10. F: [
F: ~ ~ >
as we shall see,
Here
then
Cip_oF[1,F ]
S(i) = A i e ~i'
f: Fl(A1) --> A o
so objects are exactly maps
and
f: Fl(A1) --> F2(A2)
in
It is easily checked that morphisms of sections are the usual
commutative diagrams. A similar calculation shows that a subequalizer of G
as in 1,7.10, has as objects an object
morphism
f: G A - - >
1,7.12.4
Eilenber~-Moore qate~ories.
and let c~ triples. cells by
= op~
FA
,
--> t
and
together with a
with the obvious notion of morphism.
Let ~
so that 2-functors from
We denote the 1-cells of ~: 1
A E mA
F
and
~: t
2
C~by m>
221
t.
tn Let
be as in 1.7.]1.9. C~to
Cat
are
and the generating 2T:C~__>
Cat
be a
1,7
2-functor.
57
We shall show that the cartesian quasi-limit of
the category of Eilenberg-Moore algebras of the triple consider the category by
S(*) = A e A,
where
where
4n: TnA --> A.
shows that if all
r[l,T].
~n'S
A section
~(*) = A ,
~.
S: c7~--> [I,~}
and morphisms
are determined by
A, 4.
then
[1,~S
42 = 4 T(4).
Since
S
is
For, is given
S(t n) = (id,4n)
The formula for composition in
4 = F(t): T A - - >
T
w
(7.27)
etc., so that
has to be a 2-functor, the
diagrams
T2A
~A
-> TA
A
~A
\/ A
A
must commute.
Hence
.> TA
(4: TA --> A)
is a ~-algebra°
Morphisms of
sections are clearly the same as morphisms of algebras. Replacing
~
by
A_°p
yields the Eilenberg-Moore category
of coalgebras of the corresponding cotriple. that the cartesian quasi-colimit of
~
is uninteresting (it is
as is the cartesian quasi-limit of a functor
1,7.12.5.
G:~-->
~).
Cat.
Associated fibrations As in 1,7.11.10,
F: A - - >
One easily verifies
B.
let
~: ~ - - >
Cat
have as its value
It is easily checked that
Cart q - l( l m ¢ = Z . [ I , F ]
since a section of
[I,F] --> ~
=
(F.B)
is described by a morphism
FA --> B,
while a natural transformation of sections is the same as a morphism in the indicated comma category.
Thus
222
(F,B)
has three different
1,7
58
universal properties; first, it is the pullback in
(F,B)
>
A
second,
(F,B) --> B
Cat
.>B
is the universal opfibration associated to
F
and hence left adJoint to an inclusion functor, and third it satisfies the right adJointness property of a cartesian quasi-llmit
X
(F,B)-
\
.
/
The dual construction is given by
ZI.F°P]
°p = (B,F)
223
.
1,7
1,7.13
Quasi-fibrations.
1,2.9,
there are various notions
difference case
is the use of
Definition.
the notion of 2-fibrations, of quasi-fibrations;
Fun~
(in dual form to 1,2.9)
1,7.13.1
instead
of
C --. 2
P:~
quasi-opfibration
--> B
as in
the essential
We treat here the
that will be used in the next
A 2-functor
is called a Cartesian L,
Besides
59
between
if there exists
section.
2-categories a 2-functor
as illustrated,
PunP i
iF
Fun(s )
having
S
60
as a right Cat-adjoint If
(E,f: P E - - >
is a morphism
in
~
B)
satisfying
'> B
and
SL = id.
is an object
in
the following
[P,B] universal
then
L(E,f)
mapping
property: L(E,f) E
.~ f.E
n
kz-" E'
PE
f
(7.45)
-> E"
m
> B
PE'
> PE"
.....
Pm
224
1,7
for any
m: E' --> E"
(Pn,~,g)
in
in
with
Fun~
c: n --> n'
Fun B
in ~
given
as illustrated,
n: E m >
in ~
and a 2-cell
identities, given
E
E'
L(E,Pn)
and a map
in E
(Pn',~',g'),
(n',k',h') taking
(n,k,h)
given a 2-cell
(Pc,T): (Pn,~,g) m >
In particular,
n: E - - >
in ~
Similarly,
(o,~): (n,~,h) m >
Fun P(o,~) = (Po,~).
E'
there is a unique map
Fun P(n,k,h) = (Pn,~,g).
there is a unique 2-cell with
,
60
m
and
in
Fun g
, there is a unique diagram
....> (Pn).E
(7.46) E,
with
P(~) = id,
then taking
P(An) = id.
m = l(pn,). E
and
Similarly, g = i,
given a 2-cell
o: n m >
and using the fact above
about 2-cells, there is a unique diagram
E
\
~
T~I ~
~
h c (7.47)
; \J where
P(ho) = 1pE ,,
Zx
P(Aa) : P(o),
and
P(~) : 11
. PE'
225
n',
1,7
On the other hand, taking
61
m = L(E',f')
for
f: PE' m >
B'
shows that there are uniquely defined 1-cells and 2-cells in the diagram n E' n v
L(E'¢)
L(E,f)
(7.48)
f . E ~ f . E '
making
f.: p-l(B) m >
p-l(B, )
a quasi-natural transformation.
a 2-functor and Here
P-l(B)
L(-,f): JB m >
JB, f.
is the fibre over
B;
i.e., O-cells, 1-cells and 2-cells mapping to
B,
JB:
Similarly, if
P-l(B) --> ~
v: f --> f'
is the inclusion 2-functor.
is a 2-cell in
B,
and
then there are uniquely defined
1-cells and 2-cells in the diagram
L(E,f)
E
~fE
(7.~9)
> fiE
n
L(~'#)
~
E'
E'
L(n, f ' ) / / / ~
Y L(~' ,f' )
226
T.n~/,
f.E'
1,7
making
T.: f. m >
f~
62
a quasi-natural transformation and
L(-,T)
a
modification in the diagram L(-,f)
JB
> JB.f.
(7.5o) JBf~
In particular,
this gives a functor
(-)$: B(B,B') --> Fun(p-I(B),p-I(B '))
for each pair of objects in kinds of functors from
B
B.
Globalized
to 2-Cat®
statements about various
are too complicated to go into
here. A choice of and M:
p,: ~ ~>
~. m > ~ ' i) li)
B
L
is called a cleava6e for
P.
are cartesian quasi-opfibrations,
If
P: S --> B
a 2-functor
which satisfies P'M = P L'(M,l,1) = Fun(M)L
is called cleavage preserving.
If
(i.e., L
M
"commutes with
L)
is chosen (when possible)
so
that i) ii) then
P
L(id) = id L(f.E,g) o L(E,f) = L(E,gf)
together with
opfibration.
L
is called a split-normal cartesian quasi-
We denote the 2-category of such with cleavage
preserving morphisms and Cat-natural transformations
227
over
B
by
1,7
63
Cart q-Split (B)o-
1,7.13.2 Proposition.
The inclusion
Cart q-Split (B)o ¢ I > [°P2-cat,B]
has a strict left quasi-adJoint
Proof.
We verify the conditions of
is a 2-functor and that i)
Let
PF: KF'B] --> B given
¢.
(FA h >
t
and
F: ~ - - >
B
1,7.8.R, showing that the adJoint
s
are identities.
be a 2-functor.
Then the projection
is a split-normal cartesian quasi-opfibration; for, B) ~ IF,B]
and
B f> 1
FA L(h,f):
h
B,
1
in
P,
define
> FA
i B
f
B
,,
>
f
B'
with the obvious extension to 1-cells and 2-cells in required conditions are easily verified. li)
Define
~F: F - - >
PF
Define
[PF,B].
The
¢(H) = PF"
to be the map
t%,
/~
JF
•> I F , B ]
B
where
~F
is as in (7.6').
mapping property.
For, let
Then
~F
satisfies the required universal
(P: ~ --> B,L)
228
be a split-normal
1,7
64
cartesian quasi-opfibration and consider a map
(M,m): F - - >
P
in
[°P2-Cat ,B]
~F
"> PF
IF,B]" '~/P (7.46) B
We must first construct P~ = m.
M
and
m
as illustrated with
P~ = PF
and
These are given by the diagram M
JF !
(7.47)
[ ,B] ~ > where
m
[
,B]
(M,I,I)
[ ,B] T
~
is given by
mA = PMA
mA
is the composition of the bottom row of (7.47),and
: = ~: s ( M , : , : ) ~
Thus
~A
.
is the unique (by (7.45)) morphism in a diagram
229
(7.48)
~
1,7
NA
1
11
.>NA
"> ~(~HA)
L(MA, m A ) in ~
which projects to
m A.
Since
a commutative diagram with the m
is obviously Cat-natural,
Note that
if
m = id
then
65
L(MA,mA)
in place of
same p r o j e c t i o n ,
~A
gives
~A = L(NA'mA)"
Since
it follows from (7.48) that
~ is also.
~ = id.
Now suppose that there is given another
~F
F
>
PF
(N,id)
Define
T:
(h: F A - - >
~-->
N
to be the transformation
B) E IF,B]
is the unique morphlsm in a diagram in
L(MA,hm A)
MA
> ~(h)
/
N(1FA) ~ N(1, where
whose component
(l,l,h) :
I/
\FA /
> h
The diagram
230
f
~.
N(h)
Th
vh
at
1,7
m
,.;~,~
~ ~
66
~(1FA )
,.
)
N
over the diagram
in
A (
PMAI
in
B
mA
~\
F A / .... 1
> \ FA
shows that
V~A
as required.
mA
:
VlFA
L(MA'mA)
:
nA
Furthermore, the components
are uniquely VlFA
determined by this.
Finally the diagram
MA
L
> ~(~A)
MA ,
L
> ~[(h) '
TIFA
Th
over
231
~N(1HA)
~' N(h)
1,7 PMA
mA
; FA
PMA
>
1
Th
~ B
and show that
1
is determined by
ill)
Finally, t
and
7 FA
B
hm A shows that
67
TIF A.
we must verify the conditions of 1,7.8.2
s
are
identities.
It is clear, either
directly, or by verifying the universal mapping property, (M,m): F - - >
F',
then the induced map is [m,B]
~(M,m):
(M,I,I)
IF,B]
-> [F'M,B]
which defines a 2-functor from By definition, Ep
and
tp
that if
are the
~
if
K°P2-cat,B]
P: ~ - - >
and
~ [F',B]
~
B
(7.49)
to Cart q-Split
(B)o.
is such an opfibration then
in the diagram
[P,B]
4-Y
(7.5o)
~
B
Since the
m
here is
l,
tp = 1
and (7.47), it is immediate that The map ~F
~F
and
Ep = 81L.
~ = ap ~(M,m)
is a commutative triangle,
and
is the identity, which by (7.16) implies that
s
is in fact natural.
From this, using
following result.
232
(7.49)
~ = tp [R ~(M,m)"
so the natural transformation
we observe that, since maps between opfibratlons L's,
From this,
s = 1.
Finally,
commute with the 1,7.7, we get the
1,7 I,Z.13.3 Corollary:
68
In the induced ordinary adJunction
(I,7.5),
S
> [[°P2-cat,B],I]
[F,Cart q-Split Bo] T we have i) ii)
T
is a bihomomorphism
TS = id.
In the ordinary case, the associated opfibration has another important property
(I,l.ll) which also has an analogue in this
situation.
1,7.13.4.
Propositio n .
//
There is a diagram
% > [F,B]
and
PFQF = F
[F,B]
P
strict quasi-right adJoint to
is the projection of and
PQF = I.
Define
FAI =
on
(1,e)
B) e [F,B]
B for
(f,e,g):
233
h ~>
h'
in
~.
= id: ~--> PQF
by the formulas
for (h: FA m >
e(f,e,g)
[F,B]
QF"
IF,P].
1,7
69
One verifies immediately that
e
is a strict quasi-adJunctlon.
One can also check the universal
mapping properties
(I,7.8.2) for
is quasi-natural and that
e
and
~,
(e,~)
both of which are non-
trivial. AdJoint functor theorems based on this will be discussed elsewhere. Finally,
in the discussion of Karl extensions as outlinedin
I,l.13, there is one other important aspect of fibrations which also has an analogue in this case.
1,7.13.5 proposition.
Let
Cartesian quasi-opfibration Cat®-enriched
(P: ~ - - >
B,L)
be a split-normal
with small fibres.
Then there is a 2-
imbedding
J: ~--> s[2-Ca%~ ]eProof:
If
B s B,
let
P-l(B)
be the fibre over
B
1-cells and 2-cells which project to the identity of
JB:
P-l(B) --> ~
of
s[2-Cat@,S]®.
P'I(B)
be the inclusion 2-functor, If
in
B
and
B)
and let
regarded as an object n: E ~ >
E'
in
then there is a unique diagram
E
L(E,f)
nl
such that L
>
f.E
L(n,f/i E,
that
f: B --> C
(i.e., objects,
L(E,, f)
P(f.n) = B
and
(7.5l)
f.n
" > f.E,
P(n,f) = f.
is a 2-functor in 1,7.13.1)
morphism
234
The uniqueness
(or the fact
shows that this gives rise to a
1,7
p-l(B )
70
f*
> P-1 (C)
(7.52)
~'~ L(-,f) / JB ~ / ~ - - ~ / J c
E in
s[2-CaB,~ ]®,
where
f.
is a 2-functor and
quasi-natural transformation.
L(-,f)
is a
Finally, if f
B
75 _..~c f,
is a 2-cell in
B,
then %here is a unique diagram
L(E,f)
~
f.E
E
(hT)E L(E,f' )
such that
P((~)E) = C
and
f~E
P((e )E) = 7.
this gives a 2-cell
(7.53)
One checks easily that
f.
P-I(B) ~
~
kLC-,f,,,,l,,f~
P-I(c)
/
E in
s[2-Ca~]®
modification.
in which
~
is Cat-natural and
A composition of 2-cells in
B
07
gives rise to diagrams
like (7.17) and (7.18), except that the arrow labeled is the identity since the
~'s
is a
~n
in (7.18)
are identities (i.e., in (7.19),
235
1,7
the arrow labeled
~'~n
71
is the identity.)
One checks that the
diagram like (7.19) commutes in this case by using the universal mapping property.
The details are routine.
We note that if the
fibration is not split-normal, there is still a construction as above, but what it yields is a quasi-enriched functor.
1,7.13.6
Remark:
PF: [F,B] m > [FCC].
B
Given
F:
~ m>
B,
it is easily checked that
is also an ordinary Cat-enriched opfibration, as in
This construction, which is central to the discussion of
quasi-Kan extensions, For, let
also plays a role in ordinary Kan extensions.
I: Cat---> CaR
be an inclusion of the category of small
categories into some category of big categories
(e.g., for a higher
universe), and consider P
Cat
Cat.
is a Cat-enriched opfibration and Furthermore,
PX: [I, rxn ] --> Cat flbration.
PI
[I,~-~]
of
P
for fixed
P
is a Cat-
X ¢ Cat, the restriction
to the fibre of
PI
over
X
is a
The usual description of (right, as in 1,6.12) Kan-
extensions shows that
X
is cocomplete (i.e., admits Kan-extensions
for arbitrary functors between small categories) if and only if is a Cat-enriched opflbratlon. [I, ~X ~]
and a morphism
morphism in
[I, rxu ]
For, given an object
F: Px(H) = ~ - - >
over
F
A~.>
starting at
s
x
236
B
H: ~ - - >
in Cat, a H
PX X
in
cocartesian
is a diagram
1,7
72
such that given any other map of the form over
F),
there is a unique map of the form
such that of
H
(F,~): H - - >
(1,~') (F,~) = (F,~); i.e.,
along
ZF(H)
K
(i.e.,
(I,~'): ZF(H) ~ >
K
is the Kan extension
F.
I,T.14 Quasi-Kan extensions. Let and let
X
F: ~ m >
B
be a 2-functor between small 2-categories
be a 2-category which has cartesian quasi-colimits.
F* = FUn(F,X)=
If
n(B,X) - - >
then we would like a left quasi-adJoint
Fq F
is as good a sense as possible. proof in [CCS].
I F*
?
This was discussed briefly without
The general idea is to try to follow the mheme of
I,l.13, using 1,2.9, 1,7.9.2 and 1,7.13 to replace everything by the appropriate quasi-constructions.
It turns out that quasi-opfibrations
are very well behaved, while general functors are rather poorly behaved.
l,T.14.1.
Theorem.
If
P: ~
~>
B
quasi-opfibratlon with small fibres then
P*
has a left Cat-adjoint
is a split-normal Cartesian ( ~ and
ZqP,
the fibres."
237
B
need not be small),
given by "integration along
1,7
Proof:
73
The left adJoint is denoted by
ZqP
even though it is a
Cat-adJolnt since it goes between 2-categorles of the form
Fun(~,C)
rather than of the form ~ .
Define
H: ~ - - >
is a 2-functor, then
is the composition
J
B
ZqP(H)
ZqP
H~
> sr2-cab,c ] ~
Z
q
as follows.
If
> #r2-cat~],
~
Cart q-li_~X = Q
x Here
H.
denotes composition with
Cart q - lim ~X 1,7.13.5,
by
Q
(7.55)
H; i.e., abbreviating
as in 1,7.9.2, and using the notation of
[ZqP(H)](B) = Q(HJB) [ZqP(H)](f) = Q(f.,H L(-,f))
(7.56)
[ZqP(H)](~) = Q(h ,He )
In the standard case, it is evident how this is functoral in
H.
our case, we must specifically indicate how a quasi-natural transformation
~: H --> H'
gives rise (remarkably) to a quasi-
natural transformation
ZqP(*): ZqP(H) --> ZqP(H');
namely,
238
In
X
1,7
74
(7.57) ZqP($)f = Q(I,$L(_,f))
(The reader should draw the appropriate figure like (7.54)). if
s: @ - - >
~'
is a modification, then
ZqP(S)
Finally,
is the modification
with components
ZqP(S)B = Q(I,SJB)
•
This gives a 2-functor
ZqP: Fun(E ,X) --> Fun(s,X).
TO ShOW that H: ~ - - >
X,
ZqP
K: B --> I •
transformation.
is the left Cat-adJoint to and let
Then, for
~: H --> KP
P*,
let
be a quasi-natural
BeB,
~JB: H J B - - > K P JB = A K(B)
induces a unique morphism
@B:
> K(B)
QCHJ B) tJ
[F.qP(H) ] (B)
such that, whenever
n: E - - >
E,
in
239
P'I(B),
then the diagram
1,7
75
H(E)_
hE' H(E')
--
J-____~
C
in
B,
then there is a 2 - c e l l
~f
in
the diagram
Q(~JB)
Q(~.,HL(-,f))
~B
~ Q(HJC)
rpf
~:(B) .
construched as follOws.
.
. . ~:(f)
.
For each
Z
!
/,
qoC
/
>
E,
K(C)
we h a v e
~CQ(f.,EL(-,f))~E = ~ C ~ f . ~ ( E , f ) = ~f.~HL(E,f) and one easily
checks that the 2-cells
%(E,f): K ( f ) ~ --> ~f*E HL<E,f) are the components
of a ~ o d i f i c a t i o n
240
(7.59)
1,7
76
~L(-,f): K(f)~B~(-) --> PC Q(f.,HL(-,f))~(_)
and hence induce the uniquely determined 2-cell satisfying
~f~E = ~L(E,f)"
determines a modification
A modification ~: ~ m >
~,
~f , as indicatedj
s: ~ --> ~'
clearly
and by uniqueness everywhere,
the construction is natural with respect to quasi-natural transformations and modifications in both variables Conversely, if mation, let
~: H m >
%:
KP
ZqP(H) m >
K
H
and
is a quasi-natural transfor-
be the quasl-natural transformation whose
components are given as follows.
If
E e ~
, then
~E = ~PEnE ' while if
g: E - - > E,
K.
in ~
, then
(7.60) ~g
is the composed 2-cell in
the diagram
HE
H(g)
> H(E')
j H((Pg).E)
~E,
~ ( P g ~ Q(HJpE)
> Q(HJpE,) Q((Pg).,HL(-,Pg))
~PE 1 KPE
~' K P E '
241
(7.61)
77
1,7
i.e.,
~g = (~PE' ~E' H (kg)) (~pE, ~g HL (E, Pg ~(~Pg~E )
It is easily checked that
~
also clear that on objects observe that satisfies
Is quasi-natural and that WE = ~PE~E = SE"
~Pg~E = ~L(E,Pg)
P(kg) = Id (7.46).
above formula yields
and
~pE,~
~ = ~.
To calculate = ~
,
while
It is
~g, kg
Hence, by quasl-naturality of
~,
the
~g = ~g.
The transformation
~: Id
is given by taking
$
> P* (EqP)
the identity in (7.60) and (7.61); i.e., it is
the top half of (7.61).
One can show directly that it Is Cat-natural
by using a quasl-natural transformation
e: H I >
H'
to construct a
cube wlth two sides like the top of (7.61) and with the bottom given by (7.57).
1,7.14.2
Definition.
Let
F: ~ m >
B
be a 2-functor between small
2-categories and let PF ~
Fun(B,X).
is a strict quasi left adJoint to
242
1,7
78
°PF*: °PFun(B,X)
Proof:
By 1,7.4.I,
EqPF
> °PFun(~,X) .
is left Cat-adJoint to
dualization does not affect Cat-adjoints, adJoint to
Opp~.
right adJoint to
By 1,7.13.4, ~,
P
(°P~*,l)
so, by 1,7.4 and its proof,
where
s
and (7.5'), we must calculate op~.]
~Opp.
,
where
ops.
in
°P(QF)* ,
Opp.
~
and
is right
with adJunction Thus we are in
In the relevant diagrams, (7.5)
(°P~W)(opa.)(Opp~)
mations giving the adJunction between description of
is left Cat-
is a given in 1,7.4.
a situation to apply 1,7.3, il), c).
[°P(ZqPF)*
°P(EqPF)
Since weak
is left inverse, strict quasi-
inverse, strict quasi-left adJoint to morphisms
P~ .
~
and
are the Cat-natural transfo~
~qPF
and
P~.
From the
i,7.4, the second modification (as well as
the first) has as its components values of
~
for 1-cells.
Since
is Cat-natural these 2-cells are all identities and hence the conditions for a composition of quasl-adjunctions to be a quasiadJunction are satisfied.
1,7.14.4. Remarks. i) Since
is Cat-natural and since
P~
= id,
the adJunction transformation
Id
is also Cat-natural. is only quasi-natural.
> °PF*(°PEqF)
The other adJunction transformation, however, It follows that in the associated transenden~l
quasl-adJtmction (I,7.6 and 1,7.5), S
[°P(ZqF)' °PFun(Z~'iX)]
°PFun(X, Cat)
[l,-] =tl
244
1,7
80
and will construct quasi-natural transformations
which satisfy the adJunction equations as well as the equations
es =
i)
If
K: X - - >
associated opfibration (PK,-): X - - >
Cat.
Let
1
,
~
=
1.
(7.62)
Cat is a functor, then ~ ( K )
PK: [I,K] --> X SK: (PK,-) --> K
quasi-natural transformation) in
and ~ ( K )
is the is the functor
be the morphism (i.e.,
°PFun(X,Cat)
whose components are
the functors
(%)x: (PK'x) --> K(x) given as follows.
An object of
((X',x'), f,: X' --> X)
where
(PK,X)
consists of a pair
x' e K(X'), so we set
(eK)x((X',x'),f') = (Kf')x' c K(X).
A morphism in
(PK,X)
is a pair
(g,m): ((X,,x,), f,: X, --> X ) - - > ((X",x"), f": ~ ' - - >
where set
g: X' --> ~'
satisfies
f"g = f'
and
(eK)X(g,m) = (K~')m: (Kf')x' --> (Kf")x"
m: (Kg)x' ~ > in
easily verified that this is a functor and that if the diagram
245
X)
K(X).
x".
We
It is
k: X - - >
Y,
then
1,7
(PK,X) (PK "k)
81
> (PK,Y)
(~)x
(~)y K(k)
-) KY
KX
commutes, so
eK
is a natural transformation.
This defines the 1-cells describe 2-cells v: K - - >
K',
(7.63)
e~
EK:~(K
) --> K.
We must also
corresponding to quasl-natural transformations
which map as indicated.
(P ,[1,~] )
(PK'-)
+ (PK, '-) eK'
K(-)
Here
[1,T]= [1,K] ~ >
(f,m)
to
Hence
(P ,[1,v])
-> K ' ( - )
[1,K']
(f,Ty(m) • (Vf)x)
((X',Vx,(X')),f')
takes
and
((X',x'),f') ¢ (PK,X)
takes where
(X ,x) f: X m >
to
y
(g,vr,(m)Tg).
(X,Tx(X)) and
(xx(Kf))(~).
(¢) x
to
Hence an object (K'f')(Xx,(X'))
Thus we can set
= (Tf,)x,
is a natural transformation, that
°PFun(x,Cat), and that the
246
and
m: K ( f ) x - - > y.
is taken clockwise in (7.64) to
((%)x)((x,,x,),f,)
2-cell in
to
((X,,x,), f,: X' --> X)
(g,m)
and counterclockwise to
One verifies that
(7.64)
~K'S
and
e's
sT
define a
is a
82
1,7
quasi*natural transformation the definition of em = 1.
eT
Since each
E:~
that if eK
m
> i.
It is immediate from
is a natural transformation,
is natural,
eEK
l,
then
which is the first
relation in (7.62). ii) ~(F)
If
F: ~ - - >
is the functor
PF: (F'x) --> X.
X
is an object in
(F,-): X --> Cat
Let
~F
and~(F)
then
is the functor
be the morphism QF=~F
A
[°Pcat,X]
+ (F,X)
F ~
/ F
(7.65)
X
as in I,l.ll, where
~(A)
= IFA: F A - - >
FA
and
~(f)
= (f,Ff).
There is no 2-cell indicated since the triangle commutes. morphism
<M,m>: F - - > ....
F'
in
[°Pcat,x],
~F
~<M,m>
On a
is the diagram
(F, x)
(7.66)
X
where
~<M,m>
is the natural transformation whose component at
A ¢~
is the morphism
247
1,7
1
F'MA
83
-> F,MA
F' MA
.> FA mA
in
(F',X).
~F
= l,
In particular,
m = i,
then
~<M,m~ = i.
Hence
which is the second relation in (7.62). ill)
(~)F:
if
Next we show that
(F,-) --> (PF,-)
component at an object a morphism
X,
(f,g)
<M,m>: F - - > by (7.66) and
F'
X)
(F,X) --> (PF,X),
to the object
to the morphlsm is a morphism in
(~)<M,m>
= 1.
First observe that
is the natural tra~nsformation whose
((~)F)X:
(A,FA-->
e c~ o ~
is the functor taking
(l: F A - - >
((Ff,Ff),g). [°Pcat,X],
FA, F A - - >
X)
and
9h~rthermore, if
then
~<M,m>
is given
is the diagram
i.e., it is a modification whose value at
X e X
formation whose value on an object
h > X)
(A,FA
mo rphi sm
248
is a natural transin
(F,X)
is the
1,7
F'MA
mA
F,MA
!, > F'MA
F'MA
84
>FA
.>X
FA
> FA
h
mA in
h>x
(PF,,X). On t h e o t h e r
hand,
((~)F)x:
eF,x) --> (F,x)
is Just composition; i.e., it takes an object
(h: F A - - >
X', f': X' --> X)
the object
(A,f'h: F A - - >
to
Hence
(m,1).
X)
in
(F,X)
in
(PF,X)
to
and a morphlsm
(m,g,g,l)
[((~)F)X ° ((7~)X)](A'FA-> X) = ((~9;)F)X(I: F A - - >
FA, F A - - >
X) = ( A , m - - >
X)
so the first adjunctlon equation holds on objects. To calculate it on morphisms, observe that ~ morphisms in i);
e ~
[°Pcat,x]
is natural.
takes
to natural transformations and hence, by Thus we need only calculate
(e~)F' applied
to the morphism in (7.68), which gives the identity map
F'MA
1
> F,MA
X
249
1,7
iv) that
Finally, we must show that (~)K
85
~
o
~
= i.
First observe
is the commutative triangle
[I,K]
nPK
-> (PK,X)
X
and
(~)~,
for a quasi-natural transformation
T: K --> K ',
the identity, since ~
applied to a morphism gives a commutative
triangle (cf. (7.66)).
On the other hand,
(~E)K
is
is the commutative
triangle
(PK,X)
in which
(~S)K
to the object (h,(Kg')~),
is the functor taking an object
(X,(Kf')x')
¥'
v: K - - >
and a morphism
((f,~),f,h) to
K',
fT
g'
> X
> Y
then, as in (7.64),
(~s)x
transformation whose component at the object in
(PK,X)
((X~x'),f': X' -->
where one has
X'
If
,> [1,K]
is the morphism
(Ix,(Xf,)x,)
250
in
is the natural ((X,,x'),f': X' --> X) [I,K'].
Hence
1,7
86
[ (~ ~)K " (n~)K](x,x)
= (~)K((x,x),
[ (%( ~)~ ,, (n~)~](f,~)
= (~)~((f,~),f,f)
and, on morphisms,
[(~B)V
since
(~)T
= i,
l: x - - >
x) = (X,x)
= (f,~)
we have
o ~pK](X,x ) = ( ( ~ ) ~ ) ( ( X , x ) , l : X
--> x) = (1X,~l)x) = 1.
Hence the other adjunction equation holds. One can verify that the transcendental quasi-adJunction of [21] is derived from the strict quasi-adJunctlon given here by using 1,7.5.
1,7.16.
The QuasiIyoneda Lemma. This example illustrates another aspect of quasiiadJointness.
We first describe the situation in general and then prove both local and global analogues of the Yoneda lemma. The general situation is as follows: categories,
F: ~ - - >
B
and
biJections on the objects, and
U: B - - > ~
~
and
B
are 2-
are 2-functors which are
e: F U - - >
B
and
~: A ~ >
UF
are
quasi-natural transformations whose components at objects are identity morphisms.
If we identify the objects of
A
then all that is left are the components of
and ~
and
B
via ~
F
and
at morphisms,
which look like A
FUf
> A'
A
.........~
> A'
(7.69) A
.................
> A'
A
f
'> A' UFg
251
U,
1,7
where
f c B
and
g c ~.
1,7.16.1. Proposition. F
quasi
l U
ftmctors
87
~
and
~
define a quasi-adJunction
if and only if for each pair of objects
FA,A,: ~(A,A,)
> B(A,A'),
A,A,, the
UA,A,: B(A,A') m > ~ ( A , A ' )
are adJoint (in the ordinary sense) via adJunction morphisms, and
~g
as above, which are multiplicative
~f,f = e f , ~ e f
Proof.
,
ef
in the sense that
ng, g = ng,~J~g
(7.70)
Clear. If
F
and
U
are pseudo-functors
are spoiled by having the appropriate them, but the principle is the same. U: °PB m >
opt,
then
FA,A, -~
then these nice equations
~'s, ~'s, etc., inserted in Also, if one has
UA,A,,
F: °P~ m >
o~,
so the order of adjointness
is preserved. For instance, a bicategory
~
if a cartesian closed category is regarded as
with a single object, then
functor via the diagonal and Regarded as pseudo-functors Ax -
(_)A from
Ax - is a copseudo-
is a strict copseudo-functor. op~
to itself, one has
(_)A. The local quasi-Yoneda lemma is concerned with describing
the situation of 1,5.10 what follows,
~
and
(2-~Cat,B × W ) ,
i) in terms of quasi-adjoints B
are 2-categories
; resp.,
as above.
and
,(2-~Cat, ~ X B)
is the full and locally full sub 2-category of (2-~Cat, B × ~ ) ; (2-~Cat, ~ x B),
In
determined by objects of the form
252
resp.,
88
1,7
×,4.: resp.,
[B,F] --> B
where
F:
,4--> B
is a 2-functor.
taking a 2-functor
F: , 4 m >
B
[B,F]
[F,B] m > , 4
× B
The idea is that the operation
into the objects in Spans (e-cat)
> s
x,4
,
which can be thought of as the bifibration corresponding to HomB(-,F(-)): it is not.
B °p × , 4 - - >
Cat, should be full and faithful.
Rather, it is faithful onto a quasi-reflectlve
However, sub-
category in the sense made precise by the following two theorems.
1,7.16.2
Theorem (The local quasi-Yoneda lemma).
There exists a
quasi-adJunction
~
~:
°PFun(,4,B)
> °P(2-~Cat, Bv~), ,
resp.,
~
~: ' (2-~Cat, ,4 × B) ...... > Fun(,4,B) Op
such that i)
~
is a 2-functor and
ii)
~
= Id
~
(resp., ~)
is a pseudo-
functor.
objects.
(Resp., ~
on objects.) ili)
and
~,~
and
0: Id --> ~
= id
and
~,~
and
e: ~ - - >
are identities on Id
are identities
(Cf., 1,7.16.1). Furthermore,
if
T: [B,F] --> [B,G]
the following are equivalent
253
is over
B ×,4
then
1,7
89
b)
T
is a left
[B,B]-homomorphism in the sense of 1,5.9,1%
c)
T
applied to a morphism of the form h B
> FA
FA . . . . .
>
FA
FA
is a commutative square. × B; a') c')
T
~(T)
= T,
(Resp., for b') T
T: [F,B] --> [G,B]
over
is a right [B,B]-homomorphlsm;
applied to a morphism of the form
FA
FA
> FA
. . . . . . . . . . . . . . .
FA
>
B
h is a commutative square.)
Proof: i)
If
F: ~ ~ >
~'(F) = [PB,P/~]:
If
~: F --> G
B
then
[B,F]
-->
B ×,~
.
is a quasi-natural transformation,
the morphism (1,5.3,ii) and 1,5.4 c).)
254
then
Y(~)
is
1,7
[B,F]
90
[B,q)]
> [B,G]
Bx,4 in the comma category. then
~(u)
Finally, if
u: ~ --> $'
is a modification,
is the 2-cell (I,5.3 ii) and 1,5.4,d).)
tB,v~~KB,uJ ] ~~tB,G]
2-'
[B,~ ] /
~(u):
Note that
[B,u]
is a natural transformation.
of 1,5.3,ii), that
~
is a 2-functor.
It is a special case
We regard it as a 2-functor
between the weak duals in order to have the adJointness come out correctly. Conversely, define
~([B,F]
> S ×A) = r
and given a morphism
[B,F]
•>
[B,G] (7.71)
B×A
then, following the proceedure of 1,5.9, ii), let 2-functor over
~ × ~
making the diagram
255
T
be the unique
1,7
IF,F]
(F,l,l)
91
[B,~]
>
[~',a]
> [B,a] (F,I,I)
commute and define
~(T): F --> G
to be the quasi-natural transfor-
mation corresponding to
~(T) = ~
via 1,5.2, iii).
Similarly,
k: T - - >
T'
over
~: ~ m >
T'
by 1,5.6,
modification
JF
B × ~,
> [F,F]
~ IF,G] (7.7 2 )
given a natural transformation
it determines a natural transformation
iii) and
corresponding to
~(k): ~(T) --> ~(T') ~(k) = ~JF
is the
via 1,5.2, iii).
It
follows from the uniqueness part of 1,5.6 together with 1,5.1 that this defines a functor
~F,G
for fixed
F
and
G.
itself is only a copseudo-functor
[B,F]
over
B × ~,
T
> [B,G]
T'
since, given a composit~n
> [B,H]
(7.7})
there is a modification
ST,,T: ~(T'T) between quasi-natural
........> ~(T')~(T) transformations
256
from
(7.74)
F
to
H
whose component
1,7
at
A ~~
92
is the indicated 2-cell: T(IdFA)
FA
~ GA
T(idFA)
T'
FA
T(idFA)
> GA
/
T' T(idFA)
idGA\l =
)I T' (idGA
( %,, TIA/I GA
) GA
GA
since, by definition
HA
(~T)A = T(idFA), etc.
modification, observe that the modification modification
~T'
>
HA
HA
To see that this is a 7T ,
of 1,5.10 induces a
in
[F,G] ~ [G,G] ......................
> [F,G]
O
[F,G] ~ [G,H] ................
> IF,H] O
where
T' is induced by
T'(F,l,1),
by 1,5.6,i), and that
ST, T
the composition of this with
[ ~ jF,JG] : ~
> [F,G] ~ [G,G] .
There should be a proof based on 1,5.9 that copseudo-functor,
but we have not been able to find it.
observe directly that given
~':
over
B × ~,
(~,s(_ _),id)
then
T"
[B,H] ..........> [P,K]
applied to
257
is a
Instead,
is
1,7
93
T(IdFA) FA,
GA
~'~%A)
HA
yields T(idFA) ,,,
FA
>GA
T"T,~i%
(ST",T')a
I~'(IdHA)
KA
But
T"(ST,,T)A = (ST, T, T)A
and
T(idFA ) = T(T)A , etc., so
(ST- T,) A T(T) A - (S~,T, T)A = ~(T~' )A( ST, T)A • (ST.,T,T) A
which shows that
~
is a copseudo-functor.
We choose to regard
as a pseudo-functor between the weak duals as indicated. By deflnition~ Furthermore, T~ = id .
~
and
?
are mutual inverses on objects.
it is evident that on 1-cells and 2-cells, To show that
T qu-~asl ~
a quasi-natural transformation
~: Id
258
as indicated, > ~
as well
we must describe
whose components are
1,7
94
natural transformations as illustrated T
[B,F]
over
S × ~,
nT
satisfying
Now by the definition of
~''~[B,G]
~(~) = id = ~(~T) and "qT'T ~(T) and by 1,5.~, c), ~(T)
= ~T,~T". is given by
the composition
I~JF [B,F] ~ [B,F] ~
> [B,F] ~ [F,F]
~(T)
[B,F] ~[F,G] [B,G]
Hence there is a diagram (from 1,5.10)
[B,F]
[s,F]
1 x JF> [~,F] )~ IF,F]
,[B,;]
[ "~,F) ~ (F,I,Z) zxT
[B,B] ~ [B,F] !
/ / .¢
¢ -'-->//B, G ]
(I,I,F)~ (F,
/ [B,~] ~ [B,G]
259
1,7
where
95
~ T = 7T [(!'I'F) ~ (F,I,I)], and we set
(7.75)
~T = IT (I × JF)(=): T --> ~)(T)
Since ~T
7T
at
is a natural transformation,
(h: B - - >
FA) e [B,F]
so is
~T"
The component of
is easily calculated to be the
morphlsm
B ..............B (~T)h :
~ (n
T(h) GA
in which
(~T) h
>B
FA ,~T(idFA ) > GA
GA
is the indicated 2-cell:
> FA
T FA
By definition h = idFA,
> FA
FA
=
T(IdFA
> FA
GA
~(~T) A = (~T)idFA = id,
~ GA
since, if in the above square,
then the left side is an identity morphism so the right
side is also. To derive the other equation, observe first that the equivalence of a) and c) in ill) is immediate and that
~T
is the
identity if and only if the square on the right above is commutative, with if
(~T) h = id. (~T) h = id
This certainly follows if b), ~T = id. for all
h,
then applying
260
T
Conversely,
to the diagram
1,7
B'
m
>
96
B
FA 1 FA
> FA 1
and using (m,h) E
(~T) h = id = (~T)hm
[B,B] ~
Furthermore,
[B,F].
~(~)
i
~
FA
shows that
(TT)m, h = id
for all
Hence the conditions in iii) are equivalent.
= id
is evident since
~W(~) = id
follows from
1,5.10. Finally, 1,7.16.1.
we must show that
is multiplicative
as in
This can be done using 1,5.9, but the requisite diagram
is quite complicated. shown:
~
In terms of components,
the following must be
given a composition as in (7.73), then, taking account of the
pseudo-functorlallty
of
~
in (7.70),
(~T')(~T)
i.e., given
= ~(ST',T)
(h: B ~ > FA) e [B,F],
T'(idGA)(~T)h
This follows by applying
" ~T,T
then
• (~T,)T(h) = (ST, T)Ah • (~T,T)h •
T'
to the diagram
261
1,7
97
h FA
B
~
~
~
T(idFA)
T(IdFA)
GA
(n-T)~
T(h)
GA
GA
idGA
k GA
GA
and observing that
~(T)(h) = B
etc., so that
h > FA T(idFA)
~$(T)(idFA ) = T(idFA),
T'(~T) h = (~T,T) h .
> GA
~(ST, T)A = (ST, T)Ah
and
This finishes the proof of the local quasi-
Yoneda lemma.
1,7.16.3 Theorem. above for each
~
(The global quasi-Yoneda lemma). and
B
-~: 2-Cat®
are parts of a quasi-functor
>
Spans (2-Cat)
between 2-Cat@-categories.
Proof:
The functors
Define
262
1,7
98
= A
~(F: ~ ~>
B) = ~ < - -
[B,F]
> B
[B,F]~.. ~ [8,
~(~: F --> G) = ~ ' ~ ' ~
B
[s,Q] -~u: ~ ~ >
~ , ) = [B,u]
We must describe what happens to compositions in 2-Cat®
F > B
Given
.
G> C
then there is a 2-functor
q~F,G: [B,F] ~ [~ ,G] - - . > giving a 2-cell in
Spans (2-Cat)(~,~)
[~ ,GF] from
Y(G)~(F)
to
~(GF).
For instance,
~F,G(B ~ >
FA, C ~ >
GB) = C ~ >
GB ~ >
This is clearly compatible with associativity.
GFA,
etc.
The interesting part
of the structure occurs for a pair of quasi-natural transformations o: F --> F',
~: G --> G' .
This gives rise to a commutative cube
(which is compatible with everything else).
263
1,7
99
[B,F] ~ [C,Q]
•>
[ C ,GF] . . . . .
> [ C ,GF, ]
1
Y
[B,~'] ~ [ c ,a, ]
> [B,F, ] ~ [ e ,a, ]
/ ~,F
[¢
Here
,F t
,G,F]
> [C , a , F , ]
~,F
(f: B - - >
is the natural transformation whose component at
FA,
-~>
C
[B,F'] ~ [ ~ ,a]
g: C --> GB) is the morphism
VB > G'B GB ......
g > GB
Gf
G'f
> GFA
> G'FA)
* ~ G'FA~ ~FA
in
[~ ,G'F]~ and
~F,a = id.
This is the structure required by
1,4.25. The dual situation is given by the quasl-functor
~: °P2-cat~P
> Spans (2-Cat)
where
~(F: 24 - - > B) = B
A
¢(u: ~ --> ~,] = [u,s] . [G,s]
264
1,7
i00
1,7.17. Globallzed ad~unctlan morphisms In this last example, we want to examine the correspondence in 1,5.10,il) from the stand point of 2-Cat®-categories adJoints.
In what follows,
functors,
F: ~ - - >
B
and
and quasi-
U: B --> ~
are 2-
and we consider mainly the correspondence between quasi-
natural transformations T: [~,U] --> IF,B] local result,
8: FU --> B
over
~ × B.
and 2-functors
At first we consider only a very
since we cannot describe anything llke 1,7.16.2 here.
1,7.17.1. Proposition.
There exist adJoint functors
(between
categories)
~-----iX: [Fun(B,B)](FU,B)
--> ( 2 - C a t ~ 4 ×
B)([~,U],[F,B])
resp.,
Z--~:
(2-Cat,~ ×
B)([F,B],[~,U])
--> [Fun(~/~)](~,UF)
such that i)
~=
ii)
If
id;
resp., ~ =
T: [~,U] m >
id
[F,B]
is over
× B
then the
following are equivalent:
a) ~
b) of monolds
T F.,
c)
T
is a left homomorphism with respect to the change in the sense of 1,5.10. applied to a morphism of the form
265
1,7
f
A
i01
>UB
I
UB
,>IIB UB
is a commutative × B; c') T'
a')
square.
:~(T')
(Resp.,
= T',
for
b') T'
[~,U]
over
is a right U.-homomorphism,
applied to a square of the form
FA
FA
> FA
FA - - . >
is a commutative
Proof:
T': IF,B] m >
B
square.)
We treat only the first case.
functor taking
s: F U - - >
B
o F.: [/~,U] ~ >
=
over
=
u: e m >
e'
~ × B
to
---(u) = [F,u] o F..
- - >
1,5.2 that
•
_
~E=
and
[u,u]
is the functor taking
> [FU,B].)
$: T m >
j_ are clearly functors, "
identity•
On
to
and taking a natural transformation
= ( T ) = Ju~
is the
[F,B]
the other hand, as in 1,7.6 and 1,5(5.33), ~ T: [~,U] --> IF,B]
ii), ~
into
T =~:'(e) = [ F , E ]
and taking a modification
As in 1,5.10,
T'
over
~ x B
to
and it follows from
Thus it sufficies to describe a natural
266
1,7
transformation
~: Id m > l ~
proof of 1,7.16.2, and
102
such that
~=
(7.75), one defines
~T = ~T (1 × Ju )(=): T - - > Z ~ ( T )
id = ~
.
As in the
~T = ~T [(l'l'U) XU (U,l,1)] and verifies the equations as
is done there.
1,7.17.3 Definition. i)
~
consist of the following data :
O-cells are e-categories
ii) where
Let
a 1-cell from
F: ~ m >
B
and
B
to
U: B --> ~
~
~,B, etc. is a triple
(F,~,U): B -->
are 2-functors and
~: F U - - >
B
is a quasi-natural transformation. iii) R-cells are triples o: G - - >
F
and
x: U --> V
u: G(oU) --> P(Gx)
(o,u,x):
(F,~,U) --> (G,6,V)
are quasl-natural transformations
is a modification,
where and
as indicated
GU
FU
GV
B
iv) 3-cells are pairs (s,t): (o,u,~) --> (o',u',~') s: o - - >
o'
and
t: v - - >
~'
are modifications
such that
u' " (~(sU)) = (P(Gt)) - u.
Compositions are defined as follows: a)
1-cells
(F,~,U): C m >
B
and
(H,~,W):
composition
267
B w>
~
have as
where
1,7
zo3
(~, V(HaW),WU): C--.->A b)
The weak composition of 2-cells is indicated by the diagram,
where
(c,u,x): (F,~,U) m >
(G,8,V)
and
(~,v,v): (G,~,V) --> (H,?,W).
HU
GU
HV
\
i.e., (~,v,v) c)
•
(c,u,v)= (o~, ~Hv)
8(~)
•
•
u(~U),v~)
The composition of 3-cells is given by
(s',t')(s,t) = (s's,t't)
d) The strong composition of 1-cells and 2-cells is given as follows: Consider (F,~,U)
C
(a,u,~) (G,B,V)
(H,~,W)
~
(~,v,p) (K,6,X)
268
A
(7.77)
1,7
104
Then
(H,v,W)(o,u,~)
= (oH,
u.~, w~)
(~,v,p)(F,=,u)
= (F~, =(FvU),pU)
where u.v = 6(G~T)
• u(C~U)
• =(°~u)
as illustrated. GHWU
FKWU
U FHWU
GU
GHWV
FU
FHWU
FKXU
GV
Is G
1,7.17.4. Proposition. Proof:
~
has the structure of a 2-Cat®-category.
In the situation illustrated in (7.77), there is a diagram
(FH .....WU) • (?T,v,p)(F,u,U)
~
> (GH.....WV) (O~T'P~I
(FK.....XU)
(?T,V,p)(G,~,V)
> (GK.....XV)
(K, 8,X) (F,c~,U)
.> (K,5,X) (G,~B,V) (K, 5,X) (o,u,T)
269
1,7
105
which is the required kind of structure.
We omit the lengthy
verification that this satisfies the conditions for a 2-Cat~-category.
1,7.17.5. Theorem.
There is a quasi-natural transformation between
quasi-functors on 2-Cat®-categories,
as indicated:
Prl
.> °P2_cat® °p
/ 2-Cat@
Proof:
Here
,> Spans (2-Cat)
~r
Prl(F,~,U ) = F, etc., and
components of
~
Pr2(F,G,U ) = U, etc.
The
are given as follows:
~=~
IF,S] and given
(o,u,~):
(F,~,U) --> (G,8,V)
there is a 2-cell (a Cat-
natural transformation)
[A,'r]
~,u
> EA,v]
~(o,u,~) [F,s
> [O,B]
J
[o,s]
270
1,7
whose component at in
(f: A m >
106
UB) ¢ [~,U]
is the composed morphism
[G,B] i
GA
)
I
GA
> GA
°AI FA
GUB
> GUB
FUB
> FUB
GVB
B
1
)
B
1
>
B
The main step in showing that this is quasi-natural
reduces to
showing that given (7.77), the cube
[A,wv]
[.4.W,]
[.,'4,~]
[A'Pu] / / I--(~(H~W)) [,4,xu]
___(_1 > F4, xv]
[.4.X~] [o,H,~]
> [aK,a]
[FK,S] [ oK,B] commutes.
This follows by an explicite
calculation
which is too
large to include here. There is a dual treatment ~: A - - >
UF.
However,
of quasi-natural
since the variances
are different
not seem to be any way to combine the two cases.
271
transformations there does
-
Table
Listed
of symbols
in o r d e r o f o c c u r r e n c e .
in c h a p t e r
(x) y
denotes
x.
(I) 1
i
i(-)i
(1) 1
D
(1) 1,6 (1) 1 , 6
o Cat
(i)i,
eCat
(1) I
G
(1) 2
1_
(!)3
2_
(i) 3
3_
(i)4
4_
(1)4
n
(i)4
Cat t
(I) 5
{...}
(1) 7 (1) 7
u
(I)7 V A V
,u,e,e
(I)8
A(-,-)
(2)1,6
~O, 8 1 , 8 1 , ~1
(2)2
(-)
(2) 9
o
LD
(2) 9
L G
(2) 9
L ~
(2)9
o
272
(2) 5
page y
1
-
(2)9
2-Cat
(2)9 Fun O (~, A)
(2)13
Fun
(~5,i%)
(2) 13,
Fun
(',~,%!
(4) I
(2) 13
A,A')
iso
(2) 14
[FI,F 2 ]
(2)14,
Fun
(2)15,19,
[I,F]
(2)15
XOp,°pA,opA
(2) 16
[FI,F2] 3
(2) 17
Bicat
(3)5
Pseud (~, ~')
(3)8,
Bim
(3)8
Bim (~)
(3)9
Bim (Spans %)
(3)II
g-Fun ( Axe, $) n g n - F u n ( i=l ~ A.l'~)
(4)5 (4) 16
(4) i9 u ,d,e ,u,d,e
(4) 25
Fun d ( A, ~)
(4) 25
Fun
(A,~)
(4)25
Fun u (A,~)
(4)25
g-Funx~y,z(& x ~,~:)
(4) 28
Iso-Fun ( ~k,~5)
(4)41
e
273
(5)1
(4) 1
(4)29
2
-
2-Cat®
(4) 42
(V I ,W,V 2)
(5) 2
[~,~]
(5)7
[u,v]
(5)8
(Vl, ~! ~ V2, ~2)
(5)II
W*
(5) 18
cat-ki~
(6)4
Cat-l~_~D_
(6)4
cat2-1~
(6) 4
Cat2-1~D
(6)4
Ad-Fun (A)
(6) Ii
~1
(6) 11
~1
Cart q-lim__~~
(7) 23
Cart q-lira ~
(7) 23
Cart q-li_m~ -Iso . q{ I o Cart q - l ~ _ i d ~, o
(7) 23
N
(7) 24,41
[I,F]
(7)53
Cart q-Split
~
(7) 23
(7)63
o
E q PF
(7) 72
E
(7)77
q
F
(7) 102
274
3
-
1
Index
Note : (x)y denotes page y in chapter x. adjoint morphism
(6)I
adjoint square
(6)9
associated fibration
(7)57
associated cofibration
(7)52
bicategory
(3) 1
bimodule
(3)8
canonical induced morphism
(6)21
Cartesion
(2) 16
Cartesion quasi limit
(7)23
Cartesion quasi opfibration -split normal
(3) 1 5
(7)59 (7)62
Cat-ad joint
(6)3
Cat-limit
(1) 2,
Cat2-1imit
(6)4
Cat-natural transformation
(2)7
Cat-representable
(2)7
functor
,
categorical comprehension scheme
(7)79
2-category
(2) 1
3-category
(2) 16
(6)4
Cells
(2)1
Cleavage
(7)62
Cleavage preserving
(3)15, (7)62
Cocartesion
(2) 15
At-cocomplete
(6)25
A'-cocontinuous
(6)25
Comma category
(1) 11, (7)32
Comma 2-category
(2) 20
2-comma category
(2) 14.
3-comma category
(2) 17
275
-
commutative
-
(2) 4
cube
compatible
2
pair of 2-cells
(6) 10
(6)14
conjugate copseudo
(3)5
functor
(6)29
directed discrete
(1) 1
category
double
category
(2) 18
double
functor
(2)19
double
natural
transformation
(2) 19
dual Kan extension
(6)29
Eilenberg
(7)56
- Moore categories
explicit
(5)11
formulas
exponentiation
(1)1,
fibration
(1) 12,
2-fibration
(2)19
3-fibration
(2)19
final
(6)28
2-functor
(2)5
global
adjunction
global
quasi-Yoneda
homomorphic
morphism
(1) 15,
(7) 97
pseudo-functor
(3) 5,
(3) 13
(3) 13
(3) 16
composition the fibres
(2) 3,
(2) 18
(7) 72
integration
along
interchange
law
(2) 2,
of limits
(6) 26,27
interchange
(2)8
(7) 100
lemma
homomorphism horizontal
(2) 5,
iso-
(4) 41
iso-quasi-homomorphism
(3) 16
l-quasi l-weak
(7) 3 (V) 3
Kan extension
(1) 14,
Kleisli
(7) 5O
categories
276
(2)3,
(6) 19
(2) 18
-
locally discrete
(2) I, (2)4
locally faithful
(2)7
locally full
(2)7
locally P
(2)4,7
local quasi-Yoneda lemma
(7) 88
3
-
modification
(2)8,(3)8,(4)5,16,28
multiplicative
(7)87
multiplicative category
(3)8
n-tuple category
(2)19
opfibration
(2) 15
over
(5)15
pseudo-functor
(3) 2
preserves
(6)22
quasi-adjunction
(7)3
quasi-adjunction for bicategories (7)4 quasi-colimit
(7)23
quasl-enriched functor
(4)42
quasi-fibration
(7)59
quasi-functor of two variables
(4)2
quasi-functor of n-variables
(4) 15
qu a s id- fu nc tor
(4)27
quasl -functor e quasl u -functor quasi-homomorphism
(4) 28
quasi-Kan extension
(7)72, 77
quasi-limit
(7)23
(4) 27 (3) 16
quasi-natural transformation for 2-functors
(2) 10
for pseudo-functors
(3)6
for quasi-functors
(4) 4, 16
for 2-Cat® -functors quasid-natural transformations
(4)45 (4)25
quasi
(4)28
y,z
-natural transformations
277
-
quasi-opfibration
(7) 59
quasi-Yoneda lemma
(7) 86
sections
(7) 52
set of components
(i) i
set of morphisms
(1)6
spans
(3) 9
split-normal fibration
(2)5
split-normal 2-fibration
(2) 21
split-normal 3-fibration
(2) 21
split-normal quas i- fibration
(7)62
strict pseudo-functor
(3) 5
strong codomain
(2) 2
strong composition
(2) 1
strong domain
(2)2
strong dual
(2)4
subequalizer
(7) 33
total category
(2) I
transcendental quasi-adjunction
(7) 12
transpose
(6) iO
triple category
(2) !9
triple functor
(2)19
trivial connected groupoid
(1)2
underlying category
(2)I
universal mapping property
(7)15
vertical composition
(2)3, 18
weak codomain
(2) 2
weak composition
(2)Ij 18
weak domain
(2) 2
weak dual
(2)4
Yoneda
(1) 11, (7) 86
278
4
References
[Bc]
Benabou,J. Introduction to Bicategories, Rep. Midw. Cat. Sem. I, Lecture Notes in Mathematics, voi.47, (i967), Springer-Verlag, New York.
[E-K]
Eilenberg, S. and Kelly, G.M., Closed Categories, Proc. Conf. on Cat. Alg., La Jolla 1965, SpringerVerlag (1966), p.421-562.
[FCC]
Gray, J.W., Fibred and Cofibred Categories, Proc. Conf. on Cat. Alg., La Jolla 1965, Springer-Verlag (1966) , p . 2 1 - 8 3 .
[ccs]
Gray, J.W., The Categorical Comprehension Scheme, Category Theory, Homology Theory and their Applications III, Lecture Notes in Mathematics, voi.99 (1969), Springer-Verlag, New York, p.242-312.
[CCFM]
Lawvere, F.W., The Category of Categories as a Foundation for Mathematics, Proc. Conf. on Cat. Alg., La Jolla 1965, Springer-Verlag (1966) , p.l-20.
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