Category axioms

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

1. Associativity. For any composable morphisms

   $$A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D$$

   one has

   $$(h\circ g)\circ f \;=\; h\circ (g\circ f)\in \mathrm{Hom}_{\mathcal C}(A,D).$$

2. 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

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

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

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