Endomorphism

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

Definition

An endomorphism of $A$ is a morphism

The set of endomorphisms of $A$ is denoted

Algebraic structure

With composition as multiplication,

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

1. $\mathbf{Set}$: Endomorphisms of a set $X$ are just functions $X\to X$.

2. $\mathbf{Grp}$ / $\mathbf{Ab}$: 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 $k$.

3. $R\mathbf{-Mod}$: 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).