Tensor Category

In category theory, a tensor category consists of a category , an object of , a functor , and a natural isomorphism

			(1)
			(2)
			(3)

where the data are subject to the following axioms:

1. Given four objects , , , and of , the top diagram above commutes.

2. Given two objects and of , the bottom diagram above commutes.

In the above, is called the tensor product, is called the associator, is called the right unit, and is called the left unit of the tensor category. The object is referred to as the neutral element or the identity of the tensor product.

If the maps , , and are always identities, the tensor category in question is said to be strict.

A related notion is that of a tensor R-category.