Let $(X,\tau)$ denote a topological space and $\mathcal{O}$ denote a presheaf on this space with codomain $\mathbf{Set}$. We can take the category of elements of $\mathcal{O}$, which consists of a poset $\mathrm{el}(\mathcal{O}) = \{(U,f) : U \in \tau, f \in \mathcal{O}(U)\}$ together with a forgetful map $\pi : \mathrm{el}(\mathcal{O}) \rightarrow \tau$ satisfying certain properties. If $\cal O$ happens to be a sheaf, this should be reflected in the structure of $(\mathrm{el}(\mathcal{O}),\pi).$ There should consequently be a definition of sheaf like so:
Let $(X,\tau)$ denote a topological space. Then a sheaf on $X$ consists of a poset $P$ togther with a monotone map $\pi : P \rightarrow \tau$ such that the following axioms are satisfied:
(a)
(b)
(c)
(whatever)...
I'm a bit unsure what these conditions should be (even for a presheaf). We want to be able to restrict elements of $P$ to arbitrary opens, which makes me think we should view $P$ as a "$\tau$-module", by which I mean that for all opens $U \in X$ and all $f \in P$, we can form the restriction $U \cap f$ which would normally be denoted $f \restriction_U$. The usual axioms of an action hold, e.g $$X \cap f = f, \qquad U \cap (V \cap f) = (U \cap V) \cap f.$$
I'm not quite sure whether this module structure should be viewed as extra data, or whether it can be recovered from the map $\pi$. Note that we have $\pi(U \cap f) = U \cap \pi(f)$, for example.
Ideas, anyone?
Addendum. I just learned that local homeomorphisms into a space $X$ are in bijective correspondence with sheaves on $X$. This doesn't actually answer the question, but it's related.
