Let $X$ be a nice finite cellular complex (a regular CW complex or a simplicial one), equipped with a local system $\mathcal{F}$ of free rank 1 modules over some Noetherian commutative ring $R$. What might be the most natural way to define the chain complex associated with this object?

In the majority of textbooks I saw authors follow the original recipe by Steenrod, and start by choosing arbitrarily a reference point (or a leading vertex in the case of simplicial complexes) in each cell, and then twisting the boundary operator by the module authomorphism corresponding to the homotopy class of paths connecting the respective reference points (and contained within the closure of the cell considered).

I have an impression, that at least for finite cellular complexes it is possible to define the chain complex with local coefficients in a more natural way. Let us think of $\mathcal{F}$ as of a locally constant sheaf of rank 1 free modules over $R$. In the stratification of $X$ associated with the cellular structure, let $S_d$ stand for the stratum consisting of the interiors of $d$-dimensional cells. Topologically, $S_d$ is a disjoint union of a finite number of $d$-disks. Let us define $d$-chains as global sections of the inverse image of $\mathcal{F}$ with respect to the inclusion $S_d \hookrightarrow X$ (the resulting module is naturally the product of copies of $R$, but since there are only finitely many of them, we can think of it as of the direct sum). Now, for any couple of incident cells (a $d$-cell and one of its $(d-1)$-faces), there is a natural isomorphism between the modules of global sections of pullbacks of $\mathcal{F}$ to them. To get the twisted differential it suffices to use this isomorphism (instead of explicit basises in $C_d$) in the definition of the boundary operator.

Even if this construction is limited to finite cellular complexes only, this way of getting rid of arbitrariness still seem to me to be too easy. Is there any flaw in this reasoning?

PS: The condition of finiteness of the cellular complex is indeed necessary, as can be seen from an example of a real line considered as a cellular complex consisting of 1-D cells $]n, n-1[$ and 0-D cells $\{n\}$ for $n \in \mathbb{Z}$. The pullback of $\mathcal{F}$ to the union of 1-D cells is a product of countly many copies of $R$, and thus contains a non-zero constant global section, which is a non-zero cycle in the considered construction. Maybe this can be fixed by treating the local system $\mathcal{F}$ as a cosheaf?

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    $\begingroup$ There is a discussion somewhat along these lines in my algebraic topology book (available online) in Section 3.H, pages 327-336. $\endgroup$ – Allen Hatcher Jan 1 at 19:04
  • $\begingroup$ Thank you, Allen. I remember this part of your book, and both of the proposed approaches (that of tensoring the chain modules of the universal cover over the group ring of the fundamental group and the more geometric one, with singular simplexes living in the group bundle). Indeed, with either of these approaches one can avoid arbitrary choices. $\endgroup$ – p_k Jan 1 at 20:56
  • $\begingroup$ However, my question was about the right to use the sheaf machinery, which seems already to contain a good deal of the construction of the chain complex with twisted coefficients. The question originated from some research in mathematical physics, where I would like to be able to say "this global section of $\mathcal{F}$ over cells of the top dimension is a cycle". Since nobody seems following this road, I am afraid there are some gotchas I do not see. $\endgroup$ – p_k Jan 2 at 10:52
  • $\begingroup$ There is a section 12,4 on local coeficients and local systems in our book Nonabelian Algebraic Topology (groupoids.org.uk/nonab-a-t.html} but it will take a while to get used to the machinery. Many of the basic ideas go back to A.L. Blakers, and to JHC Whitehead, but are put in a new form. For example there is a tight relation between crossed complexes and the more classical chain complexes with operators. $\endgroup$ – Ronnie Brown Jan 3 at 14:31

It seems that in the case when the space $X$ comes equipped with with a cellular structure, the right way to represent local systems is through the "locally constant constructble sheaves". If the cellular complex is regular, a constructible sheaf $\mathcal{F}$ is completely characterized by its sections $\mathcal{F}(\sigma)$ over the interior of each cell $\sigma$ and the restriction homomorphisms $\rho_{\sigma \tau}:\mathcal{F}(\tau) \to \mathcal{F}(\sigma)$ for each incidence of a cell $\sigma$ and its face $\tau$ (that is, a constructible sheaf is a functor from the face-relation poset to the target category, see e.g. https://arxiv.org/pdf/1303.3255.pdf section 4.2.2). In the case of a non-regular cellular complex (e.g. a semi-simplicial one) there might be several homomorphisms for a couple of incident cells. If $\mathcal{F}$ is locally constant, all homomorphisms $\rho_{\sigma \tau}$ are isomorphisms, and one can constuct a locally constant constructible cosheaf with the same data (by using $\rho^{-1}_{\sigma \tau}$ as extension homomorphisms).

The key fact is that if the space $X$ is compact, the ordinary cellular cohomologies and homologies with local coefficients are equivalent to the cohomologies of the locally constant constructible sheaf and the homologies of the corresponding cosheaf, respectively (I am not quite sure about what happens when $X$ is not compact, but it seems that in this case one should use cohomologies with local support and Borel-Moore homologies). The recipe for the construction of the (co)chain complex is exactly the same as proposed in the question (see https://repository.upenn.edu/cgi/viewcontent.cgi?article=4722&context=edissertations section 2.2.2).

This mostly answers my question, hope this might help somebody.

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