*-Autonomous Categories With an Appendix by Po-Hsiang Chu
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752 Michael Barr
*-Autonomous Categories With an Appendix by Po-Hsiang Chu
Springer-Verlag Berlin Heidelberg New York 1979
Author
Michael Barr Department of Mathematics McGill University 805 Rue Sherbrooke Ouest Montreal, P.Q./Canada H3A 2K6
AMS Subject Classifications (1980): 18A35, 18B15, 18B30, 22B99, 46A12, 46A20, 46M05, 46M10, 46M15, 46M40 ISBN 3-540~09563-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09563-2 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in Publication Data Barr, Michael. *·Autonomous categories. (Lecture notes in mathematics ; 752) Includes bibliographies and indexes. 1. Categories (Mathematics) I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 752. OA3.L28 no. 752 [OA169]510'.8s [512'.55]79-21746 ISBN 0-0-387-09563-2
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© by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE The category of finite dimensional vector spaces over a field many interesting properties:
It is a symmetric closed monoidal(hereafter
known as autonomous) category which has an object the functor (-,K), internal Hom opposite category.
K has
into
K, with the property that
K, induces an equivalence with its
Similar remarks apply to the category of finite
dimensional (real or complex) banach spaces.
We call such a category *-autonomous.
Almost the same thing happens with finite abelian groups, except the "dualizing object",
\R /'ll or IJ./'ll , is not an object of the category.
In no case is the
category involved complete, nor is there an obvious way of extending both the closed structure and the duality to any of the completions.
In studying these
phenomena, I came on a fairly general construction which allows you to begin with one of the above categories (and some similar ones) to embed it fully into a complete and cocomplete category which admits an autonomous structure
and
which, using the original dualizing object, is *-autonomous. In an appendix, my student Po-Hsiang Chu describes a construction which embeds
~
autonomous category into a *-autonomous category.
The embedding
described is not, however, full and is completely formal. The work described here was carried out during a sabbatical leave from II
McGill University, academic year 1975-76 mostly at the Forschungsinstitut fur Mathematik der Eidgenossische Technische Hochschule, Z~rich.
For shorter periods
I was at Universitetet i Aarhus as well as l'Universite Catholique de Louvain (Louvain-la-Neuve) and I would like to thank
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