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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.

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    $\begingroup$ For the "$\tau$"-module structure, you just want $\pi$ to be a discrete Grothendieck fibration. $\endgroup$ – Pece Aug 6 '18 at 9:27
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    $\begingroup$ Why should $P$ be a poset ? I mean how do you order $(U,f)$ and $(U,g)$ for $f,g\in \mathcal{O}(U)$ ? Shouldn't $P$ be a category ? $\endgroup$ – Max Aug 6 '18 at 11:31
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    $\begingroup$ @Max, the order relation should be $$(U,f) \leq (V,g) \iff V \supseteq U \wedge f\restriction_U = g$$ if I'm not mistaken. $\endgroup$ – goblin Aug 6 '18 at 11:35
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    $\begingroup$ @goblin : ah indeed, my bad ! $\endgroup$ – Max Aug 6 '18 at 11:45
  • $\begingroup$ It seems to me that with this ordering there may be a way to recover some stuff with $\pi$ and the notions of lower bounds : $s,t\in P$ are compatible if and only if they have a lower bound $r$ such that $\pi(r) = \pi(s)\cap \pi(t)$; and so you can express the gluing axiom (it seems) $\endgroup$ – Max Aug 6 '18 at 11:52

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