, a monoidal category
(or tensor category
) is a category C
equipped with a bifunctor
- ⊗ : C × C → C
which is associative
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.
A monoidal category is a category equipped with
- a bifunctor called the tensor product or monoidal product,
- an object 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 , called associativity, with components ,
- has as left and right identity: there are two natural isomorphisms and , respectively called left and right identity, with components and .
The coherence conditions for these natural transformations follow:
- for all , , and in , the diagram
- for all and in , the diagram
It follows from these three conditions that any such diagram (i.e. a diagram whose morphisms are built using , , , 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: A1 → B1, ..., fn: An → Bn 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 □.
- Joyal, André; Street, Ross (1993). "Braided Tensor Categories". Advances in Mathematics 102, 20–78.
- Kelly, G. Max (1964). "On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc." Journal of Algebra 1, 397–402
- Kelly, G. Max (1982). Basic Concepts of Enriched Category Theory. Cambridge University Press.
- Mac Lane, Saunders (1963). "Natural Associativity and Commutativity". Rice University Studies 49, 28–46.
- Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.). New York: Springer-Verlag.