文档介绍：A Gentle Introduction to Category
Theory
— the calculational approach —
Maarten M. Fokkinga
Version of June 6, 1994
c . Fokkinga, 1992
Maarten M. Fokkinga
University of Twente, dept. INF
PO Box 217
NL 7500 AE ENSCHEDE
herlands
e-mail: ******@
Contents
0 Introduction 3
1 The main concepts 7
1a Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1b Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1c Naturality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1d Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1e Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Constructions in categories 31
2a Iso, epic, and monic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2b Initiality and finality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2c Products and Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2d Coequalisers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2e Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A More on adjointness 59
Chapters 3 and 5 of ‘Law and Order in Algorithmics’[4]
present a categorical approach to algebras. Those chap-
ters don’t use the notions of adjunction and colimit. So
you may skip Sections 1d, and 2e, and Appendix A when
you are primarily interested in reading those chapters.
1
2 CONTENTS
Chapter 0
Introduction
Aim. In these notes we present the important notions from category theory. The
intention is to provide a fairly good skill in manipulating with those concepts formally.
What you probably will not acquire from these notes is the ability to recognise the concepts
in your daily work when that differs from algorithmics, since we give only a few examples
and those are t