I have a finite category $\mathcal{C}$, along with a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathsf{GradedCommRings}$. If $F_j$ is $j$-th graded piece of $F$, then I write $H^i(\mathcal{C},F_j)$ for the $i$-th derived inverse limit of the diagram $\mathcal{C}^{\mathrm{op}} \to \mathsf{Ab}$ of abelian groups. Equivalently, it's the $i$-th sheaf cohomology of the sheaf $F_j$, where I regard $\mathcal{C}$ as the site with trivial Grothendieck topology.

I have computed the various $H^i(\mathcal{C},F_j)$. Assembling them, there should be a cup product structure $H^i(\mathcal{C},F_j) \otimes H^{i'}(\mathcal{C},F_{j'}) \to H^{i+i'}(\mathcal{C},F_{j + j'})$. I would like to compute this product structure.

The only method I'm aware of is through sheaf cohomology, involving explicit resolutions, tensor products, and total complexes (see [1]). Unfortunately, I do not have an explicit resolution of $F$ or $F \otimes F$: it seems too complicated to do by hand, especially because my $F(c)$ are typically infinitely generated. (In my computation of $H^i(\mathcal{C},F_j)$ I circumvented this by using spectral sequences but these obscure the product structure.)

I'm led to the following questions:

• Does anyone know of a more efficient method for computing cup products of presheaf cohomology / derived limits?
• If not, is there computer software that might be capable of taking over some of the tasks outline above?

[1] : R.D. Swan. Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions.