文档介绍：Reprints in Theory and Applications of Categories, No. 1, 2005, pp. 1–289.
TOPOSES, TRIPLES AND THEORIES
MICHAEL BARR AND CHARLES WELLS
Originally published as: 1984
The first author gratefully acknowledges the support he has received from the NSERC of Canada for
the last thirty seven years.
Received by the editors xxx and, in revised form, xxx.
Reprint published on xxx.
2000 Mathematics Subject Classification: 18B25, 18C15, 18C10.
Key words and phrases: Toposes, triples, theories.
1
Michael Barr
Charles Wells
Toposes, Triples
and Theories
Version , Reprinted by Theory and Applications of Categories
Copyright 2000 by Michael Barr and Charles Frederick Wells.
This version may be downloaded and printed in unmodified form for private
use only.
Michael Barr
Peter Redpath Professor Emeritus of Pure Mathematics, McGill University
mbarr@
Charles Wells
Professor Emeritus of Mathematics, Case Western Reserve University
Affiliate Scholar, Oberlin College
cwells@
To Marcia and Jane
Contents
Preface vi
1. Categories 1
1 Definition of category . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Elements and Subobjects . . . . . . . . . . . . . . . . . . . . . . . 17
5 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6 Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
10 Filtered colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
11 Notes to Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2. Toposes 63
1 Basic Ideas about Toposes . . . . . . . . .