Categories, Allegories

315 Pages·1990·10.42 MB·English

by  P.J. Freyd

CATEGORIES, ALLEGORIES
North-Holland Mathematical Library
Board of Advisory Editors:
M. Artin, H. Bass, J. Eells, W. Feit, P. J. Freyd, F. W. Gehring,
H. Halberstam, L. V. Hormander, J. H. B. Kernperman, H. A.
Lauwerier, W. A. J. Luxemburg, L. Nachbin, F. P. Peterson, I. M.
Singer and A. C. Zaanen
VOLUME 39
NORTH-HOLLAND
AMSTERDAM. NEW YORK . OXFORD . TOKYO
CATEGORIES,
ALLEGORIES
Peter J. FREYD
Andre SCEDROV
University of Pennsylvania
Philadelphia, PA 19104-6395, USA
1990
NORTH-HOLLAND
AMSTERDAM. NEW YORK OXFORD TOKYO
* *
0
ELSEVIER SCIENCE PUBLISHERS B.V., 1990
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ISBN 0 444 70368 3 (hardbound)
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Library of Congress Cataloging in Publication Data
Freyd, Peter J
Categories, allegories / Peter J. Freyd, Andre Scedrov
p. cm.--(North-Holland mathematical library; v. 39)
Includes index.
ISBN 0-444-70368-3: -- ISBN 0-444-70367-5 (pbk.):
1. Categories (Mathematics) 2. Allegories (Mathematics)
I. SCedrov, Andrej. 1955- . 11. Title. 111. Series.
QA169.F73 1989
511.3--dc20
PRINTED IN THE NETHERLANDS
To Pam and Bonnie
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PREFACE
This book is about general concepts and methods that occur throughout
mathematics and now increasingly in theoretical computer science. Our
purpose is to give a thorough introduction to categories that emphasizes
the geometric nature of the subject and explains its connections to
mathematical logic. The book is intended for that curious person who has
seen some basic topology and algebra and feels drawn to learning and
exploring more mathematics. We also include, in small print, extensive
additional material, some of which is an enticement for further study, and
some of which will be of interest even to the experts.
Part one contains a detailed treatment of the fundamentals of Geomet-
ric Logic, which combines four central ideas: natural transformations,
sheaves, adjoint functors, and topoi. Natural transformations, together
with functors and categories, were introduced by Eilenberg and Mac
Lane in the 1940’s in their work in algebraic topology and homological
algebra. The notion of sheaf, a key notion in algebraic topology, was
formulated in the late 1940’s by Lazard and developed by Cartan, and in
another form by Leray. Adjoint functors were introduced by Kan in the
late 1950’s and developed by the first author and by Lawvere in the late
1950’s and early 1960’s. Lawvere made a crucial discovery that logical
rules of inference are instances of adjoint functors. The notion of topos
arose in the work of Grothendieck and his school in algebraic geometry in
the 1950’s and early 1960’s. It was refined by Lawvere and Tierney in the
late 1960’s and in that form it presents a categorical framework for
higher-order logic and set theory.
A special feature of our approach is a general calculus of relations
presented in part two. This calculus offers another, often more amenable
framework for concepts and methods discussed in part one. Some aspects
of our calculus of relations find their origin in the relational calculi of
Peirce and Schroeder from the last century, and in the 1940’s in the work
of Tarski and others on relational algebras. We believe that the repre-
sentation theorems we discuss are an original feature of our approach.
We thank Saunders Mac Lane and Andreas Blass for their comments
on a draft of the manuscript. Any omissions or errors are, of course,
ours.
Philadelphia, 1972-1990
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CONTENTS (and introduced notions)
In the list of notions, alternate words appear in brackets.
Chapter One: CATEGORIES
1.1. Basic definitions 3
1.1 CATEGORY, morphism, source, target, composition 3
1.11 ESSENTIALLY ALGEBRAIC THEORY 3
1.12 directed equality 3
1.13 IDENTITY MORPHISM 3
1.14 MONOID 4
1.15 DISCRETE CATEGORY 4
1.17 LEFT-INVERTIBLE, RIGHT-INVERTIBLE, ISOMORPHISM,
INVERSE, GROUPOID, GROUP 5
1.18 FUNCTOR, separating functions 5
1.182 CONTRAVARIANT FUNCTOR, OPPOSITE CATEGORY,
COVARIANT FUNCTOR 6
1.1(10) ISOMORPHISM OF CATEQORIES 6
1.2. Basic examoles and constructions 7
1.2 object, proto-morphism, SOURCE-TARGET PREDICATE
[ARROW PREDICATE] 7
1.22 category of. . . , category composed of . . . 8
1.241 CATEGORY OF SETS 9
1.242 CATEGORY OF GROUPS 9
1.243 FOUNDED (one category on another), FORGETFUL FUNCTOR,
CONCRETE CATEGORY, UNDERLYING SET FUNCTOR 9
1.244 underlying set 10
1.245 PRE-ORDERING 10
1.246 group as a category, POSET 10
1.2.51 ARROW NOTATION, puncture mark 10
1.26 SLICE CATEGORY 11
1.261 category of rings, category of augmented rings 12
1.262 LOCAL HOMEOMORPHISMS, LAZARD SHEAVES 12
1.263 counter-slice category, category of pointed sets, category of pointed
spaces 12
1.27 SMALL CATEGORY, FUNCTOR CATEGORY, NATURAL
TRANSFORMATION, CONJUGATE 13
1.271 CATEGORY OF M-SETS, RIGHT A-SET 13
1.272 CAYLEY REPRESENTATION 14
1.273 LEFT A-SET 14
1.274 NATURAL EQUIVALENCE 15