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 h 3 X4 h4 Y1 g12 g13 ❇❇❇ ❇❇ ❇❇❇ Y2 g24 ❇❇❇ ❇❇ ❇❇❇ Y3 g34 Y4. 3
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