nLab endomorphism monoid object

Definition

Let $V$ be a monoidal category. In a $V$-enriched category, the hom-object $[X, X]$ of endomorphisms on an object $X$ is a monoid object in $V$, the endomorphism monoid object.

Important examples include

• $V$ is Set, where the endomorphism monoid objects are endomorphism monoids,
• $V$ is CMon, where the endomorphism monoid objects are endomorphism rigs,
• $V$ is Ab, where the endomorphism monoid objects are endomorphism rings,
• $V$ is $R$-Mod and $R$ a commutative ring, where the endomorphism monoid objects are endomorphism $R$-algebras. For $M$ a free $R$-module with finite rank, the endomorphism $R$-algebra on $M$ is isomorphic to a matrix $R$-algebra.
• $V$ is Top, where the endomorphism monoid objects are endomorphism topological monoids,

See also

• endomorphism

• endomorphism monoid, endomorphism ring, endomorphism monoid object

• endomorphism operad

• codensity monad