Definitions

Monoidal category

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor
⊗ : C × CC
which is associative (up to a natural isomorphism), and an object I which is both a left and right identity for ⊗, (again, up to natural isomorphism). The associated natural isomorphisms are subject to certain coherence conditions which ensure that all the relevant diagrams commute. Monoidal categories are, therefore, a loose categorical analog of monoids in abstract algebra.

The ordinary tensor product between vector spaces, abelian groups, R-modules, or R-algebras serves to turn the associated categories 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.

Monoidal categories have numerous applications outside of category theory proper. 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. Braided monoidal categories have applications in quantum field theory and string theory.

Formal definition

A monoidal category is a category $mathbf C$ equipped with

• a bifunctor $otimes colon mathbf Ctimesmathbf Ctomathbf C$ called the tensor product or monoidal product,
• an object $I$ called the unit object or identity object,
• three natural isomorphisms subject to certain coherence conditions expressing the fact that the tensor operation
• is associative: there is a natural isomorphism $alpha$, called associativity, with components $alpha_\left\{A,B,C\right\} colon \left(Aotimes B\right)otimes C cong Aotimes\left(Botimes C\right)$,
• has $I$ as left and right identity: there are two natural isomorphisms $lambda$ and $rho$, respectively called left and right identity, with components $lambda_A colon Iotimes Acong A$ and $rho_A colon Aotimes Icong A$.

: The coherence conditions for these natural transformations follow:

• for all $A$, $B$, $C$ and $D$ in $mathbf C$, the diagram

commutes;

• for all $A$ and $B$ in $mathbf C$, the diagram

commutes;

It follows from these three conditions that any such diagram (i.e. a diagram whose morphisms are built using $alpha$, $lambda$, $rho$, identities and tensor product) commutes: this is Mac Lane's "coherence theorem".

A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally equivalent to a strict monoidal category.

Free strict monoidal category

For every category C, the free strict monoidal category Σ(C) can be constructed as follows:

• its objects are lists (finite sequences) A1, ..., An of objects of C;
• there are arrows between two objects A1, ..., Am and B1, ..., Bn if and only if m = n, and then the arrows are lists (finite sequences) of arrows f1: A1B1, ..., fn: AnBn of C;
• the tensor product of two objects A1, ..., An and B1, ..., Bm is the concatenation A1, ..., An, B1, ..., Bm of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists.

This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad on Cat.