There is the notion of Hochschild-Mitchell homology for a $k$-linear category $\mathcal{C}$ (here $k$ can be a field). The definition is straightforward and so are some general properties, but somehow it is hard to extract from the literature what is actually known about the Hochschild-Mitchell homology in concrete cases.
Let me be more precise:
(1) Can we generally say when the homology of $\mathcal{C}$ is be concentrated in degree zero/finite-dimensional/bounded/non-zero in certain degrees?
(2) Is there a nice interpretation for the homology groups? For, say, $H_1$ or $H_2$...
(3) Can we concretely compute the homology in the case where $\mathcal{C}$ is given by modules over some algebra $A$? Maybe in terms of Hochschild homology of $A$?
Thanks for hints concerning any of the above questions.