and string theory. Formal definition A monoidal category is a category equipped with a monoidal structure. A monoidal structure consists of the following: a bifunctor called the monoidal product, or tensor product, an object called the monoidal unit, 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 (in each of three arguments , , ) isomorphism , called associator, with components , has as left and right identity: there are two natural isomorphisms and , respectively called left and right unitor, with components and . 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: for all , , and in , the pentagon diagram commutes; for all and in , the triangle diagram commutes. 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 can be regarded as 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. For example: Set, the category of sets with the Cartesian product, any particular one-element set serving as the unit. Cat, the category of small categories with the product category, where the category with one object and only its identity map is the unit. Dually, any category with finite coproducts is monoidal with the coproduct as the monoidal product and the initial object as the unit. Such a monoidal category is called cocartesian monoidal R-Mod, 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: K''-Vect, the category of vector spaces over a field K, with the one-dimensional 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 R-algebras is monoidal with the tensor product of algebras as the product and R