Monoidal category

equipped with a bifunctor that is associative up to a natural isomorphism, and an object I that 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.

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.

Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product.

The types serve as the objects, and ⊗ is the aggregate constructor.

—store the same information even though the aggregate values need not be the same.

For type product, the identity object is the unit

, so there is only one inhabitant of the type, and that is why a product with it is always isomorphic to the other operand.

For type sum, the identity object is the void type, which stores no information, and it is impossible to address an inhabitant.

The concept of monoidal category does not presume that values of such aggregate types can be taken apart; on the contrary, it provides a framework that unifies classical and quantum information theory.

They are used to define models for the multiplicative fragment of intuitionistic linear logic.

They also form the mathematical foundation for the topological order in condensed matter physics.

Braided monoidal categories have applications in quantum information, quantum field theory, and string theory.

A monoidal structure consists of the following: Note that a good way to remember how

, cancels the identity on the left, while Rho,

The coherence conditions for these natural transformations are: A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities.

, identities and tensor product) commute: this is Mac Lane's "coherence theorem".

Further, any (small) strict monoidal category can be seen as a monoid object in the category of categories Cat (equipped with the monoidal structure induced by the cartesian product).

Monoidal functors are the functors between monoidal categories that preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product.

The concept of a category C enriched in a monoidal category M replaces the notion of a set of morphisms between pairs of objects in C with the notion of an M-object of morphisms between every two objects in C. For every category C, the free strict monoidal category Σ(C) can be constructed as follows: This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad on Cat.

The reflexivity and transitivity properties of an order, defined in the traditional sense, are incorporated into the categorical structure by the identity morphism and the composition formula in C, respectively.

Introducing a monoidal structure to the preorder C involves constructing

must be unital and associative, up to isomorphism, meaning: As · is a functor, The other coherence conditions of monoidal categories are fulfilled through the preorder structure as every diagram commutes in a preorder.

The free monoid on some generating set produces a monoidal preorder, producing the semi-Thue system.

This is one of the main diagrams used to define a monoidal category; it is perhaps the most important one.
This is one of the main diagrams used to define a monoidal category; it is perhaps the most important one.
This is one of the diagrams used in the definition of a monoidal cateogory. It takes care of the case for when there is an instance of an identity between two objects.
This is one of the diagrams used in the definition of a monoidal cateogory. It takes care of the case for when there is an instance of an identity between two objects.