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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?

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I don't see a specific question here. A sheaf (of abelian groups, say) on a discrete space is essentially just a family of abelian groups indexed by the elements of the space: i.e., it is uniquely determined by its stalks, wihch can be prescribed arbitrarily. (Note also that "enough injectives / not enough injectives" is not a menu option. You either have them or you don't. The category of abelian sheaves on any space -- so certainly including discrete spaces -- has enough injectives.) –  Pete L. Clark Mar 26 '10 at 17:30
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Also, please don't be coy about whether you are asking a homework question. That will surely annoy some people. –  Pete L. Clark Mar 26 '10 at 17:31
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I'm with Pete here. This question isn't bad enough that I would close it, but I think there just isn't much to be said. Sheaves on discrete spaces come up reasonably often in various contexts, but they aren't themselves a very interesting category. I think working this stuff out is a good exercise for a grad student to get comfortable with sheaves, but as an MO question it just falls flat. –  Ben Webster Mar 26 '10 at 17:49
    
@Pete and Ben. Thanks for your suggestions. I've edited my question and perhaps I'll try to be more specific later on, but I'm afraid that the problem that it's behind my question is too specific. –  a.r. Mar 26 '10 at 19:22
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