nLab
endomorphism

Definition

An endomorphism of an object $x$ in a category $C$ is a morphism $f:x\to x$.

An endomorphism that is also an isomorphism is called an automorphism.

Properties

Given an object $x$, the endomorphisms of $x$ form a monoid under composition, the endomorphism monoid of $x$:

$${\mathrm{End}}_C(x)={\mathrm{Hom}}_C(x,x),$$End_C(x) = Hom_C(x,x) ,

which may be written $\mathrm{End}(x)$ if the category $C$ is understood. Up to equivalence, every monoid is an endomorphism monoid; see delooping.

An endomorphism monoid is a special case of a monoid structure on an end construction. Let $d:D\to C$ be a diagram in $C$, where $C$ is a monoidal category (in the case above the monoidal structure is the cartesian product and $d$ is a constant diagram from the initial category). One defines $\mathrm{End}(d)$ as an object in $C$, equipped with a natural transformation $a:\mathrm{End}(d)\otimes d\to d$ which is universal in the sense that for all objects $Z\in C$, and any natural transformation $f:Z\otimes d\to d$ there exists a unique morphism $g:Z\to \mathrm{End}(d)$ such $a\circ (g\otimes d)=f:Z\otimes d\to d$.

Proposition

If the universal object $(\mathrm{End}(d),a)$ exists then there is a unique structure of a monoid $\mu :\mathrm{End}(d)\otimes \mathrm{End}(d)\to \mathrm{End}(d)$, such that the map $a:\mathrm{End}(d)\otimes d\to d$ is an action.

Related entries

• endomorphism, automorphism

• endomorphism monoid, endomorphism ring

• endomorphism operad