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.

isone of the common, usual constructions in Algebraic Topology! $\endgroup$ – Mariano Suárez-Álvarez Mar 8 '10 at 22:36