Monoidal category

"internal product" redirects here. It is not to be confused with inner product.

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 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.

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 information, quantum field theory, and string theory.

Formal definition

A monoidal category is a category ${\displaystyle \mathbf {C} }$ equipped with

• a bifunctor ${\displaystyle \otimes \colon \mathbf {C} \times \mathbf {C} \to \mathbf {C} }$ called the tensor product or monoidal product,
• an object ${\displaystyle I}$ 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 (in each of three arguments ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$) isomorphism ${\displaystyle \alpha }$, called associator, with components ${\displaystyle \alpha _{A,B,C}\colon (A\otimes B)\otimes C\cong A\otimes (B\otimes C)}$,
• has ${\displaystyle I}$ as left and right identity: there are two natural isomorphisms ${\displaystyle \lambda }$ and ${\displaystyle \rho }$, respectively called left and right unitor, with components ${\displaystyle \lambda _{A}\colon I\otimes A\cong A}$ and ${\displaystyle \rho _{A}\colon A\otimes I\cong A}$.

The coherence conditions for these natural transformations are:

• for all ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$ and ${\displaystyle D}$ in ${\displaystyle \mathbf {C} }$, the pentagon diagram
commutes;
• for all ${\displaystyle A}$ and ${\displaystyle B}$ in ${\displaystyle \mathbf {C} }$, the triangle diagram
commutes;

It follows from these three conditions that a large class of such diagrams (i.e. diagrams whose morphisms are built using ${\displaystyle \alpha }$, ${\displaystyle \lambda }$, ${\displaystyle \rho }$, identities and tensor product) commute: this is Mac Lane's "coherence theorem". It is sometimes inaccurately stated that all such diagrams commute.[clarification needed]

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 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.

• Many monoidal categories have additional structure such as braiding, symmetry or closure: the references describe this in detail.
• Monoidal functors are the functors between monoidal categories which preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product.
• There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid. In particular, a 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).
• A monoidal category can also be seen as the category B(□, □) of a bicategory B with only one object, denoted □.
• Rigid categories are monoidal categories in which duals with nice properties exist.
• Autonomous categories (or compact closed categories) are monoidal categories in which inverses exist; they abstract the idea of FdVect, finite-dimensional vector spaces.
• Dagger symmetric monoidal categories, equipped with an extra dagger functor, abstracting the idea of FdHilb, finite-dimensional Hilbert spaces. These include the dagger compact categories.
• 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.
• Tannakian categories are monoidal categories enriched over a field which are very similar to representation categories of linear algebraic groups.
• Spherical category
• Monoidal-category action