It is well known that for a finite group $G$, the associator of the fusion category of $G$-graded $k$-vector spaces is given by an element of $H^3(G,k^*)$, up to equivalence of categories. ($k^*$ is the multiplicative group of units in $k$.)

A crucial step when showing this is the fact that in $G$-graded vector spaces, all simple objects are invertible, and therefore the tensor product of two simples is simple again. Denote the simple object generating the $g$-graded subcategory, $g \in G$, by $k_g$. Then $k_{g_1} \otimes (k_{g_2} \otimes k_{g_3}) = k_{g_1 g_2 g_3}$ is again simple, and thus is automorphisms are given by an element of $k^*$, for each triple of group elements. Two categories of $G$-graded vector spaces may be monoidally equivalent via a monoidal functor $(F, F^2, F^0)$, but if so, the coherence morphism $F^2_{g,h}\colon Fk_g \otimes Fk_h \to F(k_g \otimes k_h)$ is again given by a number in $k^*$ for every tuple $(g,h)$ of group elements, and it's in fact the coboundary for two representatives of the same cohomology class.

We might want to generalise this result and start with a based fusion ring, corresponding to the Grothendieck ring of our future fusion category, with the simples as chosen basis. For the previous example, the fusion ring is $\mathbb{C}[G]$ (the group ring), and the chosen basis is $G$ (as a subset of the group ring).

Of course, most fusion categories don't have all simples invertible. This means, e.g. that the associators and the monoidal functor coherences live in entirely different spaces, and it's not so obvious how to repeat the cohomology construction.

**Is it still possible to classify associators as elements of some cohomology? How about other data, such as pivotal structures and braidings?** Ideally, the cohomology theory could just be formulated given the based fusion ring.

Additionally, what's the relation to already known homotopical and (co)homological data typically associated to fusion categories? I.e. is such a cohomology related to the cohomology of the Brauer-Picard groupoid? In which way is its "tangent cohomology" the Davydov-Yetter cohomology (as discussed here?