In the category theory, a mathematical subdivision, a monoidal category (or tensor category) is a category C, equipped with a bifunctor ⊗ : C × C → C
which is associative ("upto" to a natural isomorphism), and an object I, which contains both a left and right identity for ⊗ (again, "upto" (to) natural isomorphism). The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all relevant diagrams commutate. Monoidal categories are thus a loose categorical analogue of the monoids in abstract algebra.
The common tensor product between vector spaces, abelse groups, R modules or R algebra serves to change the associated categories in monoidal categories. Monoidal categories can be seen as a generalization of these and other examples.
In the category theory, monoid categories can be used to define the notion of a mono object a corresponding action on the objects of the category. They are also used in the definition of an enriched category.
Monoidal categories have numerous applications beyond the actual category theory. They are used to define models for the multiplicative fragment of the intuitional linear logic. They also form the mathematical foundation for the topological order in condensed matter. Braided monoidal categories have applications in quantum field theory and snail theory.
wiki
