Category

Definition

A category $\mathcal C$ consists of:

1. A collection $\mathrm{Ob}(\mathcal C)$ of objects .
2. For each pair $X,Y \in \mathrm{Ob}(\mathcal C)$, a collection $\mathrm{Hom}_{\mathcal C}(X,Y)$ of morphisms $X \to Y$.
3. For each triple $X,Y,Z$, a composition operation $\circ : \mathrm{Hom}_{\mathcal C}(Y,Z)\times \mathrm{Hom}_{\mathcal C}(X,Y)\to \mathrm{Hom}_{\mathcal C}(X,Z), \quad (g,f)\mapsto g\circ f.$
4. For each object $X$, an identity morphism $1_X \in \mathrm{Hom}_{\mathcal C}(X,X)$.

These data satisfy the category axioms (associativity and identity/unit laws); see category axioms .

This abstracts the behavior of functions and their composition .

Examples

1. $\mathbf{Set}$: objects are sets , morphisms are functions , composition is ordinary function composition, and $1_X$ is the identity function on $X$.
2. $\mathbf{Grp}$: objects are groups, morphisms are group homomorphisms, with composition given by composing homomorphisms.
3. $\mathbf{Top}$: objects are topological spaces, morphisms are continuous maps, with composition the usual composition of continuous maps.