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I would like to know if there exists an operation in homological algebra that generalizes the notion of covering maps for abstract chain complexes (over any field or ring, or maybe just certains where it works, for any length, or maybe up to a certain length), in the spirit of chain maps, homotopic maps, mapping cones... that generalize topological operations.

Otherwise, what would be an operation between chain complexes, that resemble the most a covering operation, except the trivial $X=\tilde{X}\otimes_{K[G]} K$ of taking co-invariants, for some chain complex $\tilde{X}$ with a linear $G$ action because here $X$ is constructed given $\tilde{X}$, and in the spirit I would like an operation that "construct" $\tilde{X}$ from $X$, like one would be able to do in topology for regular CW complex $X$. Maybe this would be a process that creates a triangulated complex out of $X$ and compute its fundamental group...

Thanks

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    $\begingroup$ Do you mean anything different from (co)fibres of chain maps? Because for a chain map $f:X\to Y$ between complexes you obtain a bicomplex and then take its total complex, things like that. Moreover you have easy to work with models for all kinds of homotopy (co)limits of complexes, etc. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 12, 2023 at 9:10
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    $\begingroup$ Perhaps you're looking for Shapiro's lemma (en.wikipedia.org/wiki/Shapiro%27s_lemma)? One version says that the homology of a covering space is isomorphic to ordinary homology with coefficients in the induced representation. $\endgroup$
    – HJRW
    Commented Jul 12, 2023 at 9:53
  • $\begingroup$ @მამუკაჯიბლაძე indeed I know that the mapping cone of a map can be seen as the total complex of a double complex built from the complexes but I am not familiar with fibre of chain map, or on how it can be related to my question. $\endgroup$
    – Virgile Guemard
    Commented Jul 12, 2023 at 11:36
  • $\begingroup$ @HJRW, isn't homology with local coefficient still relying on topology since the group ring in the tensor has to be a Z[\pi_1] module. I think my question can be restated as follows: what is a choice of a local coefficient system for general chains that mimic local coefficient systems in topology. $\endgroup$
    – Virgile Guemard
    Commented Jul 12, 2023 at 11:41
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    $\begingroup$ There’s no fundamental group of an abstract chain complex over a ring, so there won’t be any generalization of covering. $\endgroup$
    – Fernando Muro
    Commented Jul 12, 2023 at 16:05

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