# Flabby representable sheaves

Mathematics Asked on January 3, 2022

Let $$S$$ be a scheme. Consider some representable moduli functor $$mathcal{M}:(Sch/S)^{op}rightarrow Set$$ represented by some scheme $$M$$. Then for each $$Vin (Sch/S)^{op}$$, let define
$$mathcal{M}^{glob}(V):=text{im}left(mathcal{M}(S)rightarrow mathcal{M}(V)right).$$
This defines a subfunctor $$mathcal{M}^{glob}subset mathcal{M}$$. I’m interested if this functor is again representable. I tried showing that this is a open or closed subfunctor, but with no succes.

In a bit more generality, consdier a sheaf representable $$mathcal{F}$$ on some site $$mathcal{C}$$ with an initial object $$X$$. Then we can define a maximal flabby sub-pre-sheaf $$mathcal{F}^{glob}$$ by defining
$$mathcal{F}^{glob}(V):=text{im}left(mathcal{F}(X)rightarrow mathcal{F}(V)right).$$
Is this a sheaf again, and is it representable?

Let $$mathcal{C}$$ be a small site (or, at least, cofinally small) and let $$textbf{Psh} (mathcal{C})$$ be the category of presheaves on $$mathcal{C}$$. There is a functor $$Gamma : textbf{Psh} (mathcal{C}) to textbf{Set}$$ represented by the terminal presheaf (which may or may not be representable in $$mathcal{C}$$, at this level of generality), and it has a left adjoint $$Delta : textbf{Set} to textbf{Psh} (mathcal{C})$$ that sends every set $$A$$ to the "constant" presheaf defined by $$(Delta A) (U) = A$$. We have a counit morphism $$epsilon_F : Delta Gamma F to F$$ for every presheaf $$F$$, and your construction is precisely the image of this morphism. Expressed this way, the failure of $$operatorname{Im} epsilon_F subseteq F$$ to be a sheaf becomes unsurprising: usually we have to sheafify the presheaf image to obtain a sheaf.

If your goal is to construct a flabby (pre)sheaf, then it would be inappropriate to sheafify $$operatorname{Im} epsilon_F$$: sheafification can destroy flabbiness. On the other hand, if we work with presheaves then representability is a rather strong condition: indeed, representable presheaves are projective, so the epimorphism $$Delta Gamma F to operatorname{Im} epsilon_F$$ would be split. But that would make $$operatorname{Im} epsilon_F$$ a retract of a constant presheaf, hence also constant – not very interesting, I think.

Finally, let me remark that the notion of flabby (pre)sheaf does not seem to be appropriate for non-localic sites. The point of flabby sheaves of modules on a topological space or locale is that they are acyclic with respect to the global sections functor, but I don't think this is true for a general site.

Answered by Zhen Lin on January 3, 2022

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