# Alternative definition of “sheaf”

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.

• For the "$\tau$"-module structure, you just want $\pi$ to be a discrete Grothendieck fibration. – Pece Aug 6 '18 at 9:27
• 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 ? – Max Aug 6 '18 at 11:31
• @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. – goblin Aug 6 '18 at 11:35
• @goblin : ah indeed, my bad ! – Max Aug 6 '18 at 11:45
• 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) – Max Aug 6 '18 at 11:52