# nLab commutative monoid

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

### In Sets

A commutative monoid is a monoid where the multiplication satisfies the commutative law:

$x y = y x.$

Alternatively, just as a monoid can be seen as a category with one object, a commutative monoid can be seen as a bicategory with one object and one morphism (or equivalently, a monoidal category with one object).

Commutative monoids with homomorphisms between them form a category of commutative monoids.

### In any symmetric monoidal category

More generally, the concept makes sense internal to any symmetric monoidal category. See at commutative monoid in a symmetric monoidal category for details.

## Examples

• An abelian group is a commutative monoid that is also a group.

• The natural numbers (together with 0) form a commutative monoid under addition.

• Every bounded semilattice is an idempotent commutative monoid, and every idempotent commutative monoid yields a semilattice, (see that entry).

1. A commutative monoid in the symmetric monoidal category of vector spaces is a commutative algebra;

2. A commutative monoid in the symmetric monoidal category of chain complexes of vector spaces is a differential graded-commutative algebra;

3. A commutative monoid in the symmetric monoidal category of chain complexes of super vector spaces is a differential graded-commutative superalgebra.

Last revised on July 27, 2018 at 05:12:49. See the history of this page for a list of all contributions to it.