In mathematics, a monoidal category (or tensor category) is a category equipped with a binary 'tensor' functor and a unit object I. The tensor operation must be associative in the sense that there is a natural isomorphism α with components ; and I must be a left and right identity in the sense that there are natural isomorphisms λ and ρ with components and respectively. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

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. ... 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. ...

must commute. It follows from these two conditions that any such diagram commutes: this is Mac Lane's "coherence theorem".

• A monoidal category may be regarded as a bicategory with one object.
• Many monoidal categories have additional structure such as braiding or symmetry: the references describe this in detail.
• There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid.
• Monoidal categories are used to define models for linear logic.

A bicategory is a concept in category theory used to extend the notion of sameness (i. ... Braided monoidal category is a mathematical concept in terms of category theory and is, as its name suggests, a monoidal category with braiding. ... 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. ... In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ...

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...

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. ...

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.