Composition of morphisms

The rule that composes morphisms in a category, generalizing function composition.

Composition of morphisms

Definition

Let $\mathcal C$ be a category . If

f : X \to Y \quad\text{and}\quad g : Y \to Z

are morphisms in $\mathcal C$ , their composition is a morphism

g\circ f : X \to Z.

Composition is required to satisfy:

Associativity: for $f:W\to X$ , $g:X\to Y$ , $h:Y\to Z$ , $h\circ(g\circ f) = (h\circ g)\circ f.$
Unit laws: using the identity morphisms $1_X$ , $f\circ 1_X = f,\qquad 1_Y\circ f = f$ whenever $f:X\to Y$ .

This abstracts composition of functions in set theory.

In $\mathbf{Set}$ , if $f:X\to Y$ and $g:Y\to Z$ are functions , then $g\circ f$ is the usual function composition.
In $\mathbf{Grp}$ , composition is composition of group homomorphisms.
In $\mathbf{Top}$ , composition is composition of continuous maps.