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Is there a description of of sheaf cohomology for the sheaf of sections of a continuous function in terms of common constructions in Algebraic Topology?

In more detail: Any sheaf on a space X can be described as the sheaf of sections of some continuous map from the étale space Y to X. In fact, the category of sheaves (of sets) on X is equivalent to the category of maps to X which are local homeomorphisms. A sheaf of Abelian groups is the same as an Abelian group object in the category of sheaves of sets, so instead of talking about cohomology of sheaves, we could talk about cohomology of an Abelian group object in the category of local homeomorphisms to X, that is, a local homeomorphism from some space Y to X such that, roughly, every fibre has an Abelian group structure where all the multiplications (of the fibres) put together form a continuous map from Y × X Y to Y.

It seems like there should be a simple description of cohomology of X with coefficient in a sheaf of Abelian groups in terms of the corresponding map Y → X that uses only usual constructions in Algebraic Topology and the (fibrewise) group structure of Y. Is there one?

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    $\begingroup$ Computing derived functors is one of the common, usual constructions in Algebraic Topology! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Mar 8, 2010 at 22:36
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    $\begingroup$ As is computing the Cech Complex. $\endgroup$
    – Charles Siegel
    Commented Mar 8, 2010 at 23:39
  • $\begingroup$ @Mariano & Charles Siegel, both: I wasn't very explicit about what kind of constructions I wanted in the description, but I meant things like homotopy classes of maps between appropriately defined spaces (for example, ordinary cohomology), and whatever you can get from them by kernels and other such operations from homological algebra. (You may mean something like this without me realizing it, in which case I would be very grateful if you enlightened me!) $\endgroup$
    – Omar Antolín-Camarena
    Commented Mar 9, 2010 at 1:58
  • $\begingroup$ @Charles Siegel: I guess I regard the simplicial Cech nerve as a construction in Algebraic Topology and you can get the usual Cech complex by Dold-Kan, but I wanted something more along the lines of "take this sort of cohomology on Y, and the map induced to cohomology on X and do this" or "build these spaces out of X and Y and take this kind of cohomology of them", etc. (I can't say precisely what description I want but I'll know it when I see it.) $\endgroup$
    – Omar Antolín-Camarena
    Commented Mar 9, 2010 at 2:06
  • $\begingroup$ @Mariano: Could you explain what you mean? Do you mean that derived functors are just the functors induced on homotopy categories by some functors on categories with weak equivalences (or more specifically, model categories)? If so (even though I didn't say), I wanted something more concrete (or "elementary") than "well, the injective resolutions you must take to compute sheaf cohomology are just cofibrant replacements in an approriate model structure". $\endgroup$
    – Omar Antolín-Camarena
    Commented Mar 9, 2010 at 2:10

2 Answers 2

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In the case that $Y = A \times X$ is an untwisted sheaf, then there is an easy description (for reasonable spaces X and top. abelian groups A (say Hausdorff, compactly generated, locally contractible)) which is proven in G. Segal "Cohomology of Topological Groups" Sym. Math. Vol IV 1970 pg. 377. From the results of that paper it follows that for $i \geq 1$,

$$H^i(X, \mathcal{O}_A) \cong [X, B^i A]$$

where this is sheaf cohomology and $[-, -]$ denotes homotopy classes of maps, and $B^iA$ is the $i^{\text{th}}$ iterated classifying space. (Note that when A is abelian, BA is again an abelian topological group).

For twisted coefficients (i.e. arbitrary Y), there is a similar description, but you must work in the over category of spaces over X.

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  • $\begingroup$ Right, I knew something like this must be true for constant coefficients (but not a precise set of hypothesis and a reference). Could you explain to me explicitly the case with arbitrary Y? Given an Abelian group object Y in {spaces over X}, to define BY as a space over X do you just form the (simplicial space over X valued) nerve of the Abelian group object $Y\to X$, and then take its realization (defined by the usual nerve and realization adjunction for the functor $\Delta \to$ {spaces over X} that sends the n-simplex to the projection $|\Delta^n|\times X \to X$)? $\endgroup$
    – Omar Antolín-Camarena
    Commented Mar 9, 2010 at 3:11
  • $\begingroup$ That sounds like it should work if all the spaces are reasonable, but there are technical pitfalls when working in the relative setting. What you want is another abelian group object $E \to X$ with an embedding as a closed sub-object $Y \hookrightarrow E$ such that E is "contractible" in the over category, i.e. $E \simeq X$ as spaces over X. Then the quotient E/Y in spaces over X will be the correct group BY. If you phrase it like this, then most (all?) of Segal's machinery should carry over. $\endgroup$
    – Chris Schommer-Pries
    Commented Mar 9, 2010 at 21:03
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I am pretty sure that you know what I'm going to say below, if it's correct, but maybe you or someone else can set me straight if I'm wrong.

Let $X$ be a space, and let $\mathcal{F}$ be a sheaf of abelian groups on $X$. Then $\mathcal{F}$ defines a functor $\mathcal{O}(X) \to \mathbf{Ab}$ from the category $\mathcal{O}(X)$ of local homeomorphisms to $X$ to the category $\mathbf{Ab}$ of abelian groups, which satisfies descent: that is, if $U \to X$ is any local homeomorphism, then $\mathcal{F}$ can be recovered in the usual way from its pull-backs to $U, U \times_X U, \dots$ (actually, you only need the first two, and no "dots").

Now, $\mathcal{F}$ also defines a functor $$\mathcal{F}: \mathcal{O}(X) \to \mathbf{Sp}$$ where $\mathbf{Sp}$ is the $\infty$-category of spectra (namely, taking values in Eilenberg-MacLane spectra in degree zero). There is a well-defined notion of a sheaf of spectra: it's one which satisfies an analogous homotopy descent condition where you take the whole cosimplicial thing for the homotopy limit rather than an equalizer (and for hypercovers rather than Cech covers).

So $\mathcal{F}$ is a sheaf of abelian groups, but it's not a sheaf of spectra. In fact, if you take the sheafification of $\mathcal{F}$ (as a sheaf of spectra), and take its homotopy groups, you get the sheaf cohomology groups of $\mathcal{F}$. If I am not mistaken, this follows from the (degenerate) descent spectral sequence: that is, to sheafify $\mathcal{F}$, you take the inverse limit of the totalizations over each hypercover, and this gives you the sheaf cohomology groups of $\mathcal{F}$ (that is, $\pi_i$ of the sheafification is $H^{-i}$ of the sheaf over that open set).

Another way to check this is to treat $\pi_i$ of the sheafification as a $\delta$-functor on sheaves of abelian groups. The main thing to check is that if you have a sheaf of injective abelian groups, then this sheafification business doesn't give you anything new. Again, this follows from the descent spectral sequence, but there's probably another way to do it.

This doesn't quite answer your question: you want to describe the higher cohomology groups of $\mathcal{F}$ in terms of its "espace etale." I can't see how to do this in terms of the present discussion: we really needed to be in a stable context, as the sheaf cohomology groups occur in negative degrees. But again, I haven't thought too much about this.

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