文档介绍：Category Theory Lecture Notes
Daniele Turi
Laboratory for Foundations puter Science
University of Edinburgh
September 1996 – December 2001
Prologue
These notes, developed over a period of six years, were written for an eighteen lectures
course in category theory. Although heavily based on Mac Lane’s Categories for the Working
Mathematician, the course was designed to be self-contained, drawing most of the examples
from category theory itself.
The course was intended for post-graduate students in puter science at the
Laboratory for Foundations puter Science, University of Edinburgh, but was attended
by a varied audience. Most sections are a reasonable account of the material presented
during the lectures, but some, most notably the sections on Lawvere theories, topoi and Kan
extensions, are little more than a collection of definitions and facts.
Contents
Introduction 1
1 Universal Problems 1
Natural Numbers in set theory and category theory . . . . . . . . . . . . . . . 1
Universals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Basic Notions 7
Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Initial and Final Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Universality 11
4 Natural Transformations and Functor Categories 14
5 Colimits 18
Coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Coequalisers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Pushouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Initial objects as universal arrows . . . . . . . . . . . . . . . . . . . . . . . . . 20
Generalised Coproducts . . . .