nLab
commutative monoid

Contents

Context

Algebra

[[!include higher algebra - contents]]

Contents

Definition

In Sets

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

xy=yx.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).

Examples of commutative monoids in a symmetric monoidal category:

  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.