 Monoidal category

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor
 ⊗ : C × C → C
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 called natural because they together satisfy certain coherence conditions which state that all the relevant diagrams commute.) Monoidal categories are, therefore, a loose categorical analog of monoids in abstract algebra.
Ordinary tensor product makes vector spaces, abelian groups, Rmodules, or Ralgebras, 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.
Contents
Formal definition
A monoidal category is a category equipped with
 a bifunctor 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 α, called associator, with components ,
 has I as left and right identity: there are two natural isomorphisms λ and ρ, respectively called left and right unitor, with components and .
The coherence conditions for these natural transformations are:
 for all A, B, C and D in , the diagram
 for all A and B in , the diagram
It follows from these three conditions that any such diagram (i.e. a diagram whose morphisms are built using α, λ, ρ, 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.
Examples
 Any category with finite products is monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a cartesian monoidal category.
 Any category with finite coproducts is monoidal with the coproduct as the monoidal product and the initial object as the unit.
 RMod, the category of modules over a commutative ring R, is a monoidal category with the tensor product of modules ⊗_{R} serving as the monoidal product and the ring R (thought of as a module over itself) serving as the unit. As special cases one has:
 KVect, the category of vector spaces over a field K, with the onedimensional vector space K serving as the unit.
 Ab, the category of abelian groups, with the group of integers Z serving as the unit.
 For any commutative ring R, the category of Ralgebras is monoidal with the tensor product of algebras as the product and R as the unit.
 The category of pointed spaces is monoidal with the smash product serving as the product and the pointed 0sphere (a twopoint discrete space) serving as the unit.
 The category of all endofunctors on a category C is a strict monoidal category with the composition of functors as the product and the identity functor as the unit.
 Boundedabove meet semilattices are strict symmetric monoidal categories: the product is meet and the identity is the top element.
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) A_{1}, ..., A_{n} of objects of C;
 there are arrows between two objects A_{1}, ..., A_{m} and B_{1}, ..., B_{n} only if m = n, and then the arrows are lists (finite sequences) of arrows f_{1}: A_{1} → B_{1}, ..., f_{n}: A_{n} → B_{n} of C;
 the tensor product of two objects A_{1}, ..., A_{n} and B_{1}, ..., B_{m} is the concatenation A_{1}, ..., A_{n}, B_{1}, ..., B_{m} 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 2monad on Cat.
See also
 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 are monoidal categories in which inverses exist.
 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 Mobject of morphisms between every two objects in C.
References
 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. London Mathematical Society Lecture Note Series No. 64. Cambridge University Press. http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf.
 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: SpringerVerlag.
Categories: Monoidal categories
Wikimedia Foundation. 2010.
Look at other dictionaries:
Closed monoidal category — In mathematics, especially in category theory, a closed monoidal category is a context where we can take tensor products of objects and also form mapping objects . A classic example is the category of sets, Set, where the tensor product of sets A … Wikipedia
Dagger symmetric monoidal category — A dagger symmetric monoidal category is a monoidal category which also possesses a dagger structure; in other words, it means that this category comes equipped not only with a tensor in the category theoretic sense but also with dagger structure… … Wikipedia
Braided monoidal category — In mathematics, a braided monoidal category is a monoidal category C equipped with a braiding; that is, there is a natural isomorphism:gamma {A,B}:Aotimes B ightarrow Botimes Afor which the following hexagonal diagrams commute (here alpha is the… … Wikipedia
Traced monoidal category — In category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback.A traced symmetric monoidal category is a symmetric monoidal category C together with a family of… … Wikipedia
Monoidal monad — In category theory, a monoidal monad (T,η,μ,m) is a monad (T,η,μ) on a monoidal category such that the functor is a lax monoidal functor with and as coherence maps, and the natu … Wikipedia
Category of rings — In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity). Like many categories in mathematics, the category of rings is… … Wikipedia
Category of vector spaces — In mathematics, especially category theory, the category K Vect has all vector spaces over a fixed field K as objects and K linear transformations as morphisms. If K is the field of real numbers, then the category is also known as Vec.Since… … Wikipedia
Category of abelian groups — In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category.The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the… … Wikipedia
Monoidal functor — In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with… … Wikipedia
Category theory — In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects and morphisms . Categories now appear in most branches of mathematics and in… … Wikipedia