文档介绍：Michael Barr
Charles Wells
Toposes, Triples
and Theories
Version
10 September 2000
Copyright 2000 by Michael Barr and Charles Frederick Wells.
This version may be downloaded and printed in unmodified form for private use
only. It is available at sci/math/wells/pub/
and as any of the files , , ,
, .
Michael Barr
Peter Redpath Professor Emeritus of Mathematics, McGill University
barr@
Charles Wells
Professor Emeritus of Mathematics, Case Western Reserve University
Affiliate Scholar, Oberlin College
charles@
To Marcia and Jane
Contents
Preface vi
1. Categories 1
Definition of category 1
Functors 11
Natural transformations 16
Elements and Subobjects 20
The Yoneda Lemma 26
Pullbacks 29
Limits 35
Colimits 48
Adjoint functors 54
Filtered colimits 67
Notes to Chapter I 71
2. Toposes 74
Basic Ideas about Toposes 74
Sheaves on a Space 78
Properties of Toposes 86
The Beck Conditions 92
Notes to Chapter 2 95
3. Triples 97
Definition and Examples 97
The Kleisli and Eilenberg-Moore Categories 103
Tripleability 109
Properties of Tripleable Functors 122
Sufficient Conditions for Tripleability 128
Morphisms of Triples 130
Adjoint Triples 135
Historical Notes on Triples 142
4. Theories 144
Sketches 145
The Ehresmann-Kennison Theorem 149
Finite-Product Theories 152
Left Exact Theories 158
Notes on Theories 170
iv
5. Properties of Toposes 173
Tripleability of P 173
Slices of Toposes 175
Logical Functors 178
Toposes are Cartesian Closed 183
Exactness Properties of Toposes 186
The Heyting Algebra Structure on Ω 193
6. Permanence Properties of Toposes 198
Topologies 198
Sheaves for a Topology 203
Sheaves form a topos 209
Left exact cotriples 211
Left e