In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid). More explicitly, a monoidal category is a category equipped with a binary functor Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In category theory, a 2-category is a category with morphisms between morphisms. It can be formally defined as a category enriched over Cat (the category of catetgories and functors, with the monoidal structure induced by the composition). ... In category theory, an n-monoid is an n-category with only one 0-cell. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

called tensor, and a unit object I.

A monoidal category must be equipped with three natural isomorphisms expressing the fact that the tensor operation should In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

• be associative: there is a natural isomorphism α, called associativity, with components

,

• have I as left and right identity: there are two natural isomorphisms λ and ρ, respectively called left and right identity, with components

and

.

These natural transformations are subject to certain coherence conditions. All the necessary conditions are implied by the following two: for all A, B, C and D in , the diagrams

and Image File history File links Pentagon coherence condition for a monoidal category {PD} Created by me, using LaTeX with Paul Taylors diagrams package. ...

must commute. It follows from these two conditions that any such diagram commutes: this is Mac Lane's "coherence theorem". Image File history File links Triangle coherence condition for a monoidal category {PD} File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Saunders Mac Lane (born 4 August 1909) is a US mathematician. ...

Monoidal categories are used to define models for the multiplicative fragment of intuitionistic linear logic. They also from the mathematical foundation for the topological order in condensed matter. In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ... // Background Although all matter is formed by atoms, matter can have very different properties and appear in very different forms, such as solid, liquid, superfluid, magnet, etc. ...

Lax monoidal categories

A monoidal category is said to be a strict monoidal category when the natural isomorphisms α, λ and ρ are identities. A lax monoidal category is a generalization of the notion of monoidal category where the natural transformations α, λ and ρ are not required to be isomorphisms. A monoidal category is sometimes called a strong monoidal cateory or a weak monoidal category to emphasize that it is not lax.

A monoidal category may be regarded as a bicategory with one object. In mathematics, a bicategory is a concept in category theory used to extend the notion of sameness (i. ...

Examples

Any category with standard categorical products and a terminal object is a monoidal category, with the categorical product as tensor product and the terminal object as identity. Also, any category with coproducts and an initial object is a monoidal category - with the coproduct as tensor product and the initial object as identity. (In both these cases, the structure is actually symmetric monoidal.) However, in many monoidal categories (such as R-Mod, given below) the tensor product is neither a categorical product nor a coproduct. In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... A symmetric monoidal category is a monoidal category which is associative up to a natural isomorphism. ...

Examples of monoidal categories, illustrating the parallelism between the category of vector spaces over a field and the category of sets, are given below.

R-Mod	Set
Given a field or commutative ring R, the category R-Mod of R-modules (in the case of a field, vector spaces) is a symmetric monoidal category with product ⊗ and identity R.	The category Set is a symmetric monoidal category with product × and identity {*}.
A unital associative algebra is an object of R-Mod together with morphisms and satisfying	A monoid is an object M together with morphisms and satisfying
and	and
.	.
A coalgebra is an object C with morphisms and satisfying	Any object of Set, S has two unique morphisms and satisfying
and	and
.	.
	In particular, ε is unique because { * } is a terminal object.

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a module is a generalization of a vector space. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Image File history File links Associativity for R-algebras. ... Image File history File links Associativity for monoids. ... Image File history File links File links The following pages link to this file: Monoidal category ... Image File history File links Identity for monoids. ... Image File history File links Associativity for R-coalgebras. ... Image File history File links Associativity for comonoids. ... Image File history File links Identity for R-coalgebras. ... Image File history File links Identity for comonoids. ...

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.
• There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid. In particular, a 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).

Braided monoidal category is a mathematical concept in terms of category theory and is, as its name suggests, a monoidal category with braiding. ... A symmetric monoidal category is a monoidal category which is associative up to a natural isomorphism. ... In mathematics, a monoidal closed category is a closed category with an associative tensor product which is adjoint to the internal homomorphism. ... In category theory, monoidal functors are the natural notion of functor between two monoidal categories. ... In mathematics, a magma in a category, or magma object, can be defined in a category with a cartesian product. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...

References

• Joyal, André; Street, Ross (1993). "Braided Tensor Categories". Advances in Mathematics 102, 20–78.
• Mac Lane, Saunders (1997), Categories for the Working Mathematician (2nd ed.). New York: Springer-Verlag.