In category theory, a tensor category in the broad sense, also called a monoidal 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.
In some contexts in representation theory, "tensor category" is reserved for a linear monoidal category with additional rigidity or the requirement that it be an Abelian category.
Johnson-Freyd et al. (2026) asked whether a semisimple category equipped with a tensor product and associativity
isomorphisms for which the pentagon commutes only
up to a scalar on simple objects
is necessarily 
-quasi-monoidal
for some group 
, meaning that the scalar depends
only on the 
-degrees
of the four objects. Etingof et al. (2026) defined
categorical 
-cocycles for 
, 3, and 4 and proved that each is pulled back from a group 
-cocycle on the universal grading
group of the underlying based ring.
The case 
answers the question affirmatively.
Etingof et al. (2026) report collaborating with ChatGPT 5.5 Pro, especially on the appendix containing this result. As of Sep. 30, 2026, independent review had not been reported.
A related notion is that of a tensor R-category.
