文档介绍：Lecture Notes in Category Theory [Draft]
8th November 2005
These notes are prepared by Mario C´accamo from a graduate course on
Category Theory given by Glynn Winskel in the fall of 1999 at BRICS, Aarhus.
Heavy use has been made of latexed notes and exercises by Thomas Mailund
and Mikkel Nygaard Hansen, under the editorial hand of Mario C´accamo and
Glynn Winskel. The course was derived from and inspired by Martin Hyland’s
Cambridge part III Mathematics course given in 1995, when notes were taken
by Cocky Hillhorst.
Contents
1 Categories, Functors and Natural Transformations 2
Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . 7
Functor Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Constructions on Categories 9
Opposite Category and Contravariance . . . . . . . . . . . . . . . 9
Product of Categories . . . . . . . . . . . . . . . . . . . . . . . . 10
Natural Transformations between Bifunctors . . . . . . . . . . . 12
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Yoneda Lemma and Universal Properties 16
Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Limits and Colimits 21
Definition of Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Examples of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Limits in Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Limits as Products and Equalisers . . . . . . . . . . . . . . . . . 27
Definition of Colimit . . . . . . . . . . . . . . . . . . . . . . . . . 28
Examples of Colimits . . . . . . . . . . . . . . . . . . . . . . . . . 29
Colimits in Set . . . . . . . .