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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 $x$:

which may be written $End(x)$ 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 $End(d)$ as an object in $C$, equipped with a natural transformation $a: End(d) \otimes d \to d$ 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 $g: Z \to End(d)$ such $a \circ (g \otimes d) = f: Z \otimes d \to d$.

Related entries

• endomorphism, automorphism

• endomorphism monoidendomorphism monoid, endomorphism ring, endomorphism monoid object

• center of a category, center of an additive category

• endomorphism operad

• endofunction

• endofunctor

• EI-category