Monoidal category

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor

which is associative up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions which ensure that all the relevant diagrams commute. In a monoidal category, analogs of usual monoids from abstract algebra can be defined using the same commutative diagrams. In fact, usual monoids are exactly the monoid objects in the monoidal category of sets with Cartesian product.

The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples.

In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action on the objects of the category. They are also used in the definition of an enriched category.