Twisted Cech cohomology

Solution 1:

Local coefficient homology is a particular case of sheaf cohomology (cohomology of a locally constant sheaf). So, even if I'm not sure I understood precisely your wishes, I think it is possible that what you're trying to do (twisting sheaf cohomology) really amounts to consider sheaf cohomology for something like a tensor product $\mathcal F \otimes \mathcal L$, where $\mathcal L$ is the locally constant sheaf corresponding to your local coefficients. If that's true, you may be happy with the genuine Čech cohomology of that particular sheaf.

Two classical references for sheaf cohomology for topologists are Iversen's Cohomology of Sheaves and Dimca's Sheaves in Topology. The latter is considerably terser than the former, but easier to find. In particular, those books deal with a general expression of Poincaré duality in sheaf cohomology (Poincaré-Verdier duality) which requires twisting (by the orientation local coefficient system $\mathcal L_{\textrm{or}}$) so they will spend some time explaining things quite close to the things you seem to dream about.

A quite ancient reference from the Séminaire Cartan by Frenkel alludes to a construction which seems related, but I must say it seems quite opaque to me.

(Even if it's only distantly related to your question, I'd like to take this opportunity to quote two books which break the omertà on local coefficients: G.W. Whitehead's Elements of homotopy theory and Davis & Kirk's Lectures on Algebraic Topology. Neither of their account on this topic is exhaustive or perfect, but at least, they do not content themselves with a two-line remark.)