Sorry if this question is too broad, but if I explain why I'm *really* interested in it, then it would be too technical and specific. So, let me try to ask it this way.

I would like to know properties of sheaves on discrete spaces. That is, let $X_{\mathrm{dis}}$ be a topological space with the discrete topology. What can be said about sheaves on $X_{\mathrm{dis}} $? -With values in, say, the category of abelian groups, complexes of an abelian category, your prefered category, an abstract category...

For instance, I've already done part of my homework, and found, that, if I'm not wrong:

- Such sheaves are determined by its values on points $\mathcal{F}(x)$: for any $U \subset X_{\mathrm{dis}}$, $\mathcal{F} (U) = \prod_{x \in U} \mathcal{F}(x)$.
- For any $x \in X_{\mathrm{dis}}$, $\mathcal{F}_x = \mathcal{F}(x)$.
- Every such a sheaf is
*flasque*(flabby). So, for instance, there is no great deal in deriving the direct images functor: for any map $f: X_{\mathrm{dis}} \longrightarrow Y\ $, $f_*$ is an exact functor.

What else can be said? Any reference for these toy sheaves?

EDIT. As I've been told, sheaves on discrete spaces appear in different contexts. For instance, you can describe the cosimplicial Godement resolution with them: let

$$ I = id : X_{\mathrm{dis}} \longrightarrow X $$

be the identity map between $X$ with the discrete topology and any topological space $X$. Then the cosimplicial Godement resolution can be defined as follows: $I$ induces a couple of adjoint functors (direct and inverse images)

$$ I_* : Sh(X_{\mathrm{dis}}) \leftrightarrows Sh(X): I^* \ , $$

you have a triple associated

$$ T = I_* I^* : Sh(X) \longrightarrow Sh(X) $$

and the cosimplicial Godement resolution may be defined, using the standard construction associated to a triple (see McLane's "Categories for the working mathematician) as

$$ C^p (\mathcal{F}) = T^{p+1} (\mathcal{F})\ . $$

Ok. Any other contexts where sheaves on discrete spaces appear?

toospecific. – a.r. Mar 26 '10 at 19:22