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, of which monoidal categories can be considered an abstraction, is that of a system of data types closed under a type constructor that takes two types and builds an aggregate type; the types are 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 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.


Monoidal preorders

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Monoidal preorders, also known as "preordered monoids", are special cases of monoidal categories. This sort of structure comes up in the theory of string rewriting systems, but it is plentiful in pure mathematics as well. For example, the set of natural numbers has both a monoid structure (using + and 0) and a preorder structure (using ≤), which together form a monoidal preorder, basically because and implies . We now present the general case.

It's well known that a preorder can be considered as a category C, such that for every two objects , there exists at most one morphism in C. If there happens to be a morphism from c to c' , we could write , but in the current section we find it more convenient to express this fact in arrow form . Because there is at most one such morphism, we never have to give it a name, such as . The reflexivity and transitivity properties of an order are respectively accounted for by the identity morphism and the composition formula in C. We write iff and , i.e. if they are isomorphic in C. Note that in a partial order, any two isomorphic objects are in fact equal.

Moving forward, suppose we want to add a monoidal structure to the preorder C. To do so means we must choose

Thus for any two objects we have an object . We must choose and to be associative and unital, up to isomorphism. This means we must have:

and .

Furthermore, the fact that · is required to be a functor means—in the present case, where C is a preorder—nothing more than the following:

if and then .

The additional coherence conditions for monoidal categories are vacuous in this case because every diagram commutes in a preorder.

Note that if C is a partial order, the above description is simplified even more, because the associativity and unitality isomorphisms becomes equalities. Another simplification occurs if we assume that the set of objects is the free monoid on a generating set . In this case we could write , where * denotes the Kleene star and the monoidal unit I stands for the empty string. If we start with a set R of generating morphisms (facts about ≤), we recover the usual notion of semi-Thue system, where R is called the "rewriting rule".

To return to our example, let N be the category whose objects are the natural numbers 0, 1, 2, ..., with a single morphism if in the usual ordering (and no morphisms from i to j otherwise), and a monoidal structure with the monoidal unit given by 0 and the monoidal multiplication given by the usual addition, . Then N is a monoidal preorder; in fact it is the one freely generated by a single object 1, and a single morphism 0 ≤ 1, where again 0 is the monoidal unit.

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.


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].