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
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