Formal Category Theory: Adjointness for 2-Categories
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re Notes in Mathematics Edited by A Dold, Heidelberg and B. Eckmann, Zurich
391 John W. Gray University of Illinois at Urbana-Champaign, Urbana, II/USA
Formal Category Theory: Adjointness for 2-Categories
Springer-Verlag Berlin· Heidelberg· NewYork 1974
AMS Subject Classifications (1970): Primary: 18005, 18025 Secondary: 18A25, 18A40 ISBN 3-540-06830-9 Springer-Verlag Berlin· Heidelberg· New York ISBN 0-387-06830-9 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under§ 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin · Heidelberg 197 4. Library of Congress Catalog Card Number 74·791 0. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Contents
Introduction !,1
Categories Yoneda
!,2
!,3
11
Adjointness.
12
Fibrations.
12
Adjoint Functor Theorem.
13
Kan ex tens ions.
14
2-Categories.
16
2-functors .
20
Cat-natural transformations
22
Quasi-natural transformations.
25
Modifications ..
28
2-comma category.
29
3-category.
31
3-comma category
32
double category ..
33
2-and 3-categorical fibrations.
35
Bicategories.
38
Pseudo-functors.
40
Quasi-natural transformations.
43
Examples.
45
Biro
45
Spans
46
x.
46
Birr. {B). Biro (Spans Fibrations.
x.l.
48 50
IV I,4
Properties of Fun(A,B} and Pseud (A,B).
55
Quasi-functor of two variables
56
Characterization of Fun(A,B)
59
Composition quasi-functor
67
Quasi-functor of n-variables
69
Tensor product
73
Quasid-natural transformations
80
Quasi -functor.
81
Monoidal closed category structure
83
Pseud
86
X
(A, B)
Appendix li· Universal copseudo-functor.
I,5
Appendix
~.Iso-Fun
Appendix
~.categories
enriched in 2-Cat
92 95 0
Properties of 2-comma categories
101
Universal property.
103
Composition
106
Explicit formulas. Functors over
v1
x
111
v2
Fibration and monoid properties.
I,6
96
115
Homomorphism properties
120 124
Examples
134
Adjoint morphisms in 2-categories
136
Examples
137
Uniqueness, composition and preservation Adjoint Squares
139 144
Examples.
152
Kan extensions
154
Examples
156
Formal criterion for adjoint
158
Cocompleteness.
160
Interchange of limits.
161
Final
163
v I,7
Quasi-adjointness
166
Definitions.
168
Uniqueness, composition and preservation.
169
Transcendental quasi-adjunction.
177
Universal mapping properties.
180
Examples.
187
Some general principles.
187
Some Fini.te quasi-limits.
197
Quasi-colimits in Cat.
201
Quasi-limits in Cat.
217
Quasi-fibrations.
224
Quasi-Kan extensions.
237
The Categorical Comprehension Scheme The Quasi-Yoneda Lemma .
244 251
Globa
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