Cohomology Ring of a small category $\mathsf{C}$

Question

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!

Answer 1

This category of functors is an abelian category with generating projective objects: the Yoneda projective $P_c$ for each object $c \in \mathcal C$ given by $P_c(d)=\mathbb Z[Hom_{\mathcal C}(c,d)]$. One forms projective resolutions and defines $Ext$ groups in the usual way. And has already been suggested, higher $lim^i$ functors are a special case. (If $\mathcal C$ only has a finite number of isomorphism classes of objects, one can sum these projective generators together to get a single small projective generators: that would be $\mathbb Z[\mathcal C]$.)

There are many many variations on this idea, and many many examples and calculations that have been studied.

Answer 2

The case of non-finite $C$ is more subtle than one would think because there are different notions of "representation" one could use.

For example in the finite case $1_{RC}$ decomposes into pairwise orthogonal idempotents $1_{RC} = \sum_{X\in Ob(C)} id_X$. Consequently every representation $V$ decomposes as a finite direct sum $V=\bigoplus_X id_X \cdot V$.

In the general case you still have the orthogonal idempotents $id_X$, but there is no a priori way to sum them. One way out of this could be to introduce a topology such that $1_{RC} = \sum_X id_X$ is a convergent series. But then one would need to restrict the attention to representations that are compatible with this topology in a suitable sense.

For example: One could define $RC$ as the projective limit of $RC_0$ over all full subcategories $C_0\subseteq C$ with $Ob(C_0)$ finite. This is a sort of "pro-finite" category-algebra and $RC$ inherits a topology by declaring all $RC_0$ to be discrete. Then with the appropriate topological notion of "representation" (I think $V$ discrete and $RC \times V\to V$ continuous or something like that should work), one has $V=\sum_{X\in C_0} id_X V$ for a finite $C_0$ (now dependent on $V$) and one can work with that. (Note that this is automatic if $V$ is finitely generated over $R$ which is the case most pure algebraists are interested in most of the days)

If $R$ is something more specialised, say $\mathbb{C}$, there are other ways one could try to topologise this category algebra and its category of representations. Probably you want $Hom(X,Y)\subseteq RC$ to have a reasonable topology first, say a banach space, and choose the topology on $RC$ in a compatible way. Once you do this though, you run into all the same difficulties that people in functional analysis have with infinite-dimensional topological algebras. In other words there are probably whole books that could be written about the subtleties of these algebras and the difference between different approaches.