Chapter 1

Categories and

Functors

The subject of algebraic topology is historically one of the first in which huge

diagrams of functions became a standard feature. The language of category

theory is intended to provide tools for understanding such diagrams, for

working with them, and for studying the relations between them. In this

chapter we begin to develop and make use of this powerful language.

1.1. Diagrams

Before getting to categories, let’s engage in an informal discussion of di-

agrams. Roughly speaking, a diagram is a collection (possibly infinite)

of ‘objects’ denoted A, B, X, Y , etc., and (labelled) ‘arrows’ between the

objects, as in the examples

A

f

h

◆◆◆◆◆◆◆◆◆◆◆◆◆

B

g

C

and

X1

f12

f13

❇❇❇

❇❇❇❇❇

h1

X2

f24

❇❇❇

❇❇❇❇❇

h2

X3

f34

h3

X4

h4

Y1

g12

g13

❇❇❇

❇❇❇❇❇

Y2

g

❇❇❇24

❇❇❇❇❇

Y3

g34

Y4.

3

http://dx.doi.org/10.1090/gsm/127/01