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' ⊗ {displaystyle otimes } is defined) such that the tensor product is symmetric (i.e. A ⊗ B {displaystyle Aotimes B} is, in a certain strict sense, naturally isomorphic to B ⊗ A {displaystyle Botimes A} for all objects A {displaystyle A} and B {displaystyle B} 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. In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a 'tensor product' ⊗ {displaystyle otimes } is defined) such that the tensor product is symmetric (i.e. A ⊗ B {displaystyle Aotimes B} is, in a certain strict sense, naturally isomorphic to B ⊗ A {displaystyle Botimes A} for all objects A {displaystyle A} and B {displaystyle B} 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. 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 s A B : A ⊗ B → B ⊗ A {displaystyle s_{AB}:Aotimes B o Botimes A} 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.