Assume that $\mathcal{C}$ is a small category and that $\mathcal{F} \in \mathsf{ob(Ab^{\mathsf{C}})}$, is a covariant functor. When our category has finitely many objects then a classical theorem from *Mitchell* allows us to identify in that case the category of repesenations of $\mathsf{C}$, $\mathsf{Ab^{\mathsf{C}}}$ with $R\mathsf{C}-\mathsf{mod}$, where by $R\mathsf{C}$ is denoted the category algebra (defined as the free $R$-module, generated by the morphisms of $\mathsf{C}$). However, in the latter case we can define the $i$-th cohomology of $\mathsf{C}$, to be $H^{i}(\mathsf{C}, M) := {Ext^i_{R\mathsf{C}}}(\underline{R}, M)$, and exploit the intuition given by $R\mathsf{C}-\mathsf{mod}$, which is just a category of modules.

However, the assumption of $\mathsf{C}$ being with finite objects is quite restrictive (mostly applied when the category is induced by a group $G$), hence I was thinking, is there any other definition with this assumption "chopped off"?

A paper by *Fei Xu - On the cohomology Rings of Small Categories*, seems to be a standard source for this material, however I wasn't able to understand the definition he provides for the cohomology, since he uses the notion of $n$-th higher inverse limit $\varprojlim^n_{\mathsf{C}} \mathcal{F}$, which doesn't seem quite familiar to me. So, if someone wants to give an alternative (provided it exists) definition, or to give me a reference/definition of this higher inverse limit is more than welcome.

P.S.

I'm familiar with the notion of higher inverse limit in general, which by definition is the right derived functor of the left exact functor $\varprojlim$, but the above higher limit is something obscure to me and haven't confronted it before.

Thank you!