I got into a discussion today about how, just as with all the other ways of computing singular homology, there should be an internal sort of integration pairing for Cech cohomology with "Cech homology", whatever it is. My sense is that probably a Cech $n$-chain should be a formal sum of $(n+1)$-fold intersections, or maybe more precisely a formal sum of sections thereover. The boundary map should be determined by $\partial U_{i_0, \ldots, i_n} = \sum_\alpha (-1)^\alpha U_{i_0,\ldots,\hat{i_\alpha},\ldots,i_n}$, except for the serious problem that there's no way to extend sections. So perhaps instead my $n$-chains really should just be formal sums of intersections (with $\mathbb{Z}$-coefficients)? The abstract-ish reason for this is that abelian groups are just $\mathbb{Z}$-modules, and perhaps this is the same story as the pairing for nonorientable manifolds, where you twist one or the other (but not both) of homology and cohomology by the orientation sheaf. But there's an asymmetry here, because the definition of a sheaf makes it so that I have no choice but to twist my cohomology instead of my homology, and that feels wrong to me.
So, what's the right story? And assuming I've more or less got the right definition, how should I relate this back to singular homology?