Category axioms - Imported Knowlpedia Content

A category $\mathcal C$ consists of:

a collection of objects $\mathrm{Ob}(\mathcal C)$ ,
for each pair $A,B\in\mathrm{Ob}(\mathcal C)$ , a set of morphisms $\mathrm{Hom}_{\mathcal C}(A,B)$ ,
a partially defined composition operation $\circ:\mathrm{Hom}_{\mathcal C}(B,C)\times \mathrm{Hom}_{\mathcal C}(A,B)\to \mathrm{Hom}_{\mathcal C}(A,C), \quad (g,f)\mapsto g\circ f,$
and for each object $A$ , an identity morphism $\mathrm{id}_A\in \mathrm{Hom}_{\mathcal C}(A,A)$ .

The category axioms are the following two laws.

Axioms

Associativity. For any composable morphisms $A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D$ one has $(h\circ g)\circ f \;=\; h\circ (g\circ f)\in \mathrm{Hom}_{\mathcal C}(A,D).$

Identity laws. For any morphism $f:A\to B$ , $\mathrm{id}_B\circ f = f \quad\text{and}\quad f\circ \mathrm{id}_A = f.$

These axioms ensure that one can unambiguously write composites like $h\circ g\circ f$ without parentheses.

Examples

$\mathbf{Set}$ . Objects are sets and morphisms are functions. Composition is ordinary function composition and identities are identity functions.

$\mathbf{Grp}$ . Objects are groups and morphisms are group homomorphisms; composition and identities are the usual ones.

$R\text{-}\mathbf{Mod}$ . Objects are left $R$ -modules and morphisms are $R$ -linear maps; composition and identities are the usual ones.