In mathematics, a monoidal category (or tensor category) is a 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.

A rather different application, for which monoidal categories can be considered an abstraction, is a system of data types closed under a type constructor that takes two types and builds an aggregate type. The types serve as the objects, and ⊗ is the aggregate constructor. The associativity up to isomorphism is then a way of expressing that different ways of aggregating the same data—such as and —store the same information even though the aggregate values need not be the same. The aggregate type may be analogous to the operation of addition (type sum) or of multiplication (type product). 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.[1]

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

Formal definition

A monoidal category is a category equipped with a monoidal structure. A monoidal structure consists of the following:

Note that a good way to remember how and act is by alliteration; Lambda, , cancels the identity on the left, while Rho, , cancels the identity on the right.

The coherence conditions for these natural transformations are:

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.

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.


Properties and associated notions

It follows from the three defining coherence conditions that a large class of diagrams (i.e. diagrams whose morphisms are built using , , , identities and tensor product) commute: this is Mac Lane's "coherence theorem". It is sometimes inaccurately stated that all such diagrams commute.

There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid from abstract algebra. Ordinary monoids are precisely the monoid objects in the cartesian monoidal category Set. 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.

Every monoidal category can be seen as the category B(∗, ∗) of a bicategory B with only one object, denoted ∗.

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.

Free strict monoidal category

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.


Preordered monoids

A preordered monoid is a monoidal category in which for every two objects , there exists at most one morphism in C. In the context of preorders, a morphism is sometimes notated . 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. If and , then the objects are isomorphic which is notated .

Introducing a monoidal structure to the preorder C involves constructing

and must be unital and associative, up to isomorphism, meaning:

and .

As · is a functor,

if and then .

The other coherence conditions of monoidal categories are fulfilled through the preorder structure as every diagram commutes in a preorder.

The natural numbers are an example of a monoidal preorder: having both a monoid structure (using + and 0) and a preorder structure (using ≤) forms a monoidal preorder as and implies .

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

See also


  1. ^ Baez, John; Stay, Mike (2011). "Physics, topology, logic and computation: a Rosetta Stone" (PDF). In Coecke, Bob (ed.). New Structures for Physics. Lecture Notes in Physics. Vol. 813. Springer. pp. 95–172. arXiv:0903.0340. CiteSeerX doi:10.1007/978-3-642-12821-9_2. ISBN 978-3-642-12821-9. ISSN 0075-8450. S2CID 115169297. Zbl 1218.81008.
  2. ^ a b Fong, Brendan; Spivak, David I. (2018-10-12). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory". arXiv:1803.05316 [math.CT].