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.
