The notion of homology is simple. One looks for p-order chain C_p and on looks for the boundary operator delta, that is applicable to the chain C_p and that defines closed and exact chains. There is operation defined on the p-chains and that means that p-chains C_p form some kind of closed structure, it may be group of more involved structure. p-order (co-)homology group is just the factor-group of closed p-chains wrt exact p-chains. I.e. exact p-chains defines factor classes in (co-)homology groups. All that is fine. Such group is denoted by H^p(M), in which M is some manifold or other structure over which the p-chains are defined. So - when we are speaking about (co-)homology, we are thinking about (co-)homology groups (factor-groups) of closed and exact p-chains (or p-forms).
But sometimes I am meeting another notion of "homology" which is described by H^p(S,T), where S, T are sets or categories. I.e. https://arxiv.org/pdf/2106.14587.pdf has formula (3.2) with notion H^0(C_+, X_+) where C_+ and X_+ are categories and H^0(..., ...) is called co-homology.
I can not understand the meaning of such notation H^p(..., ...) (with 2 arguments). Is this some kind of group? What are the notions of p-chain or p-form here and what are the notions of boundary operator? And why notation H^(..., ...) has 2 arguments? I understand that I can define p-chains or p-forms that act only on one space, why there are 2 spaces?
I have very scarce understanding of co-(homologies) - at the level of one review chapter from https://www.cambridge.org/core/books/string-theory-and-mtheory/0D112662D065F738422C8A6E507545AB and I am trying to apply this notion to the references article about toposes of deep neural networks. And H^p(..., ...) notation/notion is unrecognizable for me.