#
nLab
endomorphism monoid object

Contents
### Context

#### Algebra

#### Category theory

#### Enriched category theory

# Contents

## 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

Last revised on March 15, 2024 at 00:58:02.
See the history of this page for a list of all contributions to it.