Kan extensions in Enriched Category Theory
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Eduardo J. Dubuc University of Chicago, Chicago, ILIUSA
Kan Extensions in Enriched Category Theory
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Springer-Verlag Berlin· Heidelberg· NewYork 1970
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 1970. Library of Congress Catalog Card Number77-13n42 Printed in Germany. Title No. 3302
PRE F ACE Category Theory is rapidly coming of age as a Mathematical discipline.
In this process it now ap'-
pears that a central role will be played by the notion of an enriched category.
These categories,
with homsets in a closed category,in particular, in a symmetric monoidal closed categoryseemed initially very complex and difficult to manage effectively.
However, independent work by various
expertsYoneda, Linton,
EilenbergKelly,
Lambek, Bunge, Ulmer, Gray, Palmquist, and othershas considerably improved the situation.
A vital
step was the discovery of the proper use of tensors, cotensors, and Kan extensions for enriched categories (A discovery made simultaneously and independently by Benabou and by Kelly with Day).
As a result, an
efficient presentation of enriched categories is now possible. This paper by Dubuc collects all these ideas in a compact exposition which makes this efficiency very clearand which also serves as a basis for Dubuc's own original contributions.
I have, there-
fore, recommended to the editors of the
Notes
series the rapid publication of this paper, to provide easy access to this foundation for future development.
Saunders Mac Lane
TABLE OF CONTENTS
Introduction.
...................................... ...
Terminology ••••••••••••••••••••••••••••••••••••••••••••••
o.
V-monomorphisms.
I.
COMPLETENESS CONCEPTS
II.
V-adjunctions
.
1
1.
V-1 imi t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . .
7
2.
Tensors and cotensors •.•••..••.•••..•.•••••.••••..• 18
3.
Ends. . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . • . • • . . . . . .. 29
4.
Kan extensions..................................... 39
5.
V- Yone$ lemzna..................................... 57
V-MONADS 1. 2. 3.
Semantics-Structure (meta> Adjointness ••••••••••••• 60 Characterization of monadic V-functors ••••••••••••• 76 Clone of operations. V-codanse and V-cogenerating V- f'unc tors , • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 80
4. III.
COMPLETE CATEGORIES
1. 2. 3• IV.
Additional properties •••••••••••••••••••••••••••••• 88 V-continuous V-functors •••••••••••••••••••••••••••• lll V-complete V-categories •••••••••••••••••••••••••••• 120 Relative
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