文档介绍：Category Theory for Scientists
David I. Spivak
February 28, 2013
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arXiv: [] 27 Feb 2013
How can mathematics make this diagram meaningful?
2
Contents
1 Introduction 5
A brief history of category theory . . . . . . . . . . . . . . . . . . . . . . . 7
Intention of this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
What is requested from the student . . . . . . . . . . . . . . . . . . . . . . 10
Category theory references . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 The category of sets 13
Sets and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Ologs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Products and coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Finite limits in Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Finite colimits in Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Other notions in Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Categories and functors, without admitting it 63
Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Databases: schemas and instances . . . . . . . . . . . . . . . . . . . . . . 91
4 Basic category theo