nLab symmetric monoidal closed category

Contents

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

A symmetric monoidal closed category is

which as such is also:

  1. symmetric monoidal

  2. closed monoidal.

Properties

Relation between monoidal and strong monads

For monads on symmetric monoidal closed categories there is a close relation between structures of monoidal monads and of strong monads.

For the moment see at enriched monad – Relation to strong and monoidal monads for more on this.

Examples

Example

Any cartesian closed category is symmetric monoidal closed.

Example

A Bénabou cosmos (a good base for enrichment in enriched category theory) is defined to be a bicomplete symmetric monoidal closed category.

References

Last revised on September 2, 2023 at 15:19:54. See the history of this page for a list of all contributions to it.