Indexed Categories and Their Applications
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661 P. T. Johnstone R. Pare R. D. Rosebrugh D. Schumacher R. J. Wood G. C. Wraith
Indexed Categories and Their Applications Edited by p. 1. Johnstone and R. Pare
Springer-Verlag Berlin Heidelberg New York 1978
Editors Peter 1. Johnstone Department of Pure Mathematics University of Cambridge 16 Mill Lane Cambridge CB2 1SB/England
Robert Pare Department of Mathematics Dalhousie University Halifax, N.S. B3H 4H8/Canada
AMS Subject Classifications (1970): 18A05, 18ClO, 18D20, 18D30
ISBN 3-540-08914-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08914-4 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 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE The recent development of the theory of indexed categories has its origins in the programme, first proposed by F.W. Lawvere, of learning how to develop familiar notions .over an arbitrary base topos in order to remove their on classical set theory. This programme has already led to great advances in our understanding of continuously variable structures (sheaves, bundles, etc.) and their relation to the corresponding constant structures. However, in order to develop category theory itself in this way, it was realized that one needed to develop a notion of "family of objects (or morphisms) indexed by a variable set (= an object of the base topos)", and a suitably flexible technique for handling such indexed families. This realization was first made by Lawvere himself, who mentioned it in his lectures at Dalhousie University in 1970. In 1972-73 he developed, in unpublished notes written at Perugia, a detailed theory of families for complete categories with small horns. Independently, J. Penon was developing his theory of locally internal categories (= indexed categories with small homs) from the point of view of enriched categories. Then J. Benabou and J. Celeyrette, knowing of Penon's work but not of Lawvere's, developed their theory of families using fibrations. Their results were presented by Benabou in a series of lectures at the Seminaire de Mathematiques Superieures of the Universitt de in the summer of 1974. In 1973-74, R. Pare and D. Schumacher feit the need for a theory of indexed categories, mainly as a tool for establishing the adjoint functor theorems and using them to prove the existence of free algebras in a topos. Although they were aware of Lawvere's 1970 remarks, their own development of the theory was well advanced when they received his Perugia notes and then, a little later, atte
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