文档介绍：Category Theory
Lecture Notes
for ESSLLI
Michael Barr
Department of Mathematics and Statistics
McGill University
Charles Wells
Department of Mathematics
Case Western Reserve University
c Michael Barr and Charles Wells, 1999
Contents
Preface iv
1 Preliminaries 1
Graphs 1
Homomorphisms of graphs 2
2 Categories 4
Basic definitions 4
Functional programming languages as categories 6
Mathematical structures as categories 8
Categories of sets with structure 10
Categories of algebraic structures 11
Constructions on categories 13
3 Properties of objects and arrows 17
Isomorphisms 17
Terminal and initial objects 18
Monomorphisms and subobjects 19
Other types of arrow 22
4 Functors 26
Functors 26
Actions 30
Types of functors 32
Equivalences 34
5 Diagrams and naturality 37
Diagrams 37
Natural transformations 42
Natural transformations between functors 46
Natural transformations involving lists 47
Natural transformations of graphs 48
natural transformations and functors 49
The Yoneda Lemma and universal elements 50
6 Products and sums 55
The product of two objects in a category 55
Notation for and properties of products 57
Finite products 64
Sums 69
Deduction systems as categories 71
7 Cartesian closed categories 73
Cartesian closed categories 73
Properties of cartesian closed categories 77
Typed λ-calculus 79
λ-calculus to category and back 80
iii
iv Contents
8 Limits and colimits 82
Equalizers 82
The general concept of limit 83
Pullbacks 86
Coequalizers 88
Cocones 89
9 Adjoints 92
Free monoids 92
Adjoints 94
Further topics on adjoints 97
10 Triples 99
Triples 99
Factorizations of a triple 100
11 Toposes 102
Definition of topos 102
Properties of toposes 104
Presheaves 10