Endomorphism - Knowlpedia

Let $\mathcal C$ be a category and let $A$ be an object of $\mathcal C$ .

An endomorphism of $A$ is a morphism

f:A\longrightarrow A.

The set of endomorphisms of $A$ is denoted

\mathrm{End}_{\mathcal C}(A)\;:=\;\mathrm{Hom}_{\mathcal C}(A,A).

Algebraic structure

With composition as multiplication,

(f,g)\mapsto f\circ g,

the set $\mathrm{End}_{\mathcal C}(A)$ is a monoid with identity element the identity morphism $\mathrm{id}_A$ .

An endomorphism is an automorphism precisely when it is invertible (i.e., an isomorphism $A\to A$ ).

Examples

$\mathbf{Grp}$ / $\mathbf{Ab}$ : Endomorphisms of a group (or abelian group) $G$ are group homomorphisms $G\to G$ . For example, $n\mapsto kn$ defines an endomorphism of $\mathbb Z$ for each integer $k$ .

$R\mathbf{-Mod}$ : Endomorphisms of an $R$ -module $M$ are $R$ -linear maps $M\to M$ . For a free module $R^n$ , $\mathrm{End}_R(R^n)$ can be identified with the ring of $n\times n$ matrices over $R$ (via the action on the standard basis).