Monoidal category: Inequivalent associators Context
In my lecture notes on tensor categories it says:

"For a given category $C$ and a given tensor product $\otimes$, inequivalent associators can exist."

Questions

*

*What notion of equivalence is (usually) meant here?


*Is it simply the equality of two natural transformations?
That is, does the proposition read 'There exist monoidal categories $(C, \otimes, I, a_1, l_1, r_1)$ and $(C, \otimes, I, a_2, l_2, r_2)$ with associators $a_2 \neq a_1$'?


*What are examples of such categories with inequivalent associators?
 A: Take the category $Vect^\mathbb{Z}_\mathbb{K}$ of $\mathbb{Z}$ graded $\mathbb{K}$-vector spaces with the graded tensor product:
$$
(V\otimes W)_n=\bigoplus_{i+j = n}(V_i\otimes W_{j}).
$$
There's the usual associator:
$$
(a\otimes b)\otimes c \in(U\otimes V)\otimes W\mapsto a\otimes (b\otimes c)\in U\otimes (V\otimes W).
$$
Another associator takes the grading into account:
$$
(a\otimes b)\otimes c \in(U\otimes V)\otimes W
\mapsto
(-1)^{i+k} a\otimes (b\otimes c)\in U\otimes (V\otimes W),
$$
where $i$ and $k$ are the gradings of $a$ and $c$, respectively.
The index $j$ of $b$ was omitted for the pentagon axiom to work.
The monoidal categories defined by these associators aren't monoidally equivalent.
In fact, a function $a:\mathbb{Z}^3\to\mathbb{K}^*$ defines an associator for $Vect^\mathbb{Z}_\mathbb{K}$ iff
$$
a(r,s,t)a(r,st,v)a(s,t,v)a(r,s,tv)^{-1}a(rs,t,v)^{-1} = 1
$$
for all $r,s,t,v\in\mathbb{Z}$.
This is the same as saying that $a$ is a nontrivial 3-cocycle of $\mathbb{Z}$ with coefficients in $\mathbb{K}^*$.
For more details you can see Example 1.7 of these lectures notes.
edit: An example very similar to this one is worked out in detail at Kerodon: the monoidal structure of a 3-cocycle is defined in Example 2.1.3.3, and in Example 2.1.6.8 it's proven that 3-chains $a,a'$ define equivalent monoidal structures if and only if they are cohomologous. This is also revisited at Example 2.1.15.
