Symmetric monoidal category

In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" formula_1 is defined) such that the tensor product is symmetric (i.e. formula_2 is, in a certain strict sense, naturally isomorphic to formula_3 for all objects formula_4 and formula_5 of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field "k," using the ordinary tensor product of vector spaces. 
Definition.
A symmetric monoidal category is a monoidal category ("C", ⊗, "I") such that, for every pair "A", "B" of objects in "C", there is an isomorphism formula_6 that is natural in both "A" and "B" and such that the following diagrams commute:
In the diagrams above, "a", "l" , "r" are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.
Examples.
Some examples and non-examples of symmetric monoidal categories:
Properties.
The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an formula_12 space, so its group completion is an infinite loop space.
Specializations.
A dagger symmetric monoidal category is a symmetric monoidal category with a compatible dagger structure.
A cosmos is a complete cocomplete closed symmetric monoidal category.
Generalizations.
In a symmetric monoidal category, the natural isomorphisms formula_6 are their "own" inverses in the sense that formula_14. If we abandon this requirement (but still require that formula_2 be naturally isomorphic to formula_3), we obtain the more general notion of a braided monoidal category.