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Added to flabby sheaf several characterizations of flabbiness, an external one which, unlike the usual definition, is manifestly local, and several internal ones.
Thanks!
(Should the entry not state the definition in more generality than over sites of opens?)
It’s not obvious to me what the correct generalisation is in that case. I like Ingo’s definition (4), but the special role of subsingletons makes me wonder if this concept really makes sense in the non-localic case.
Right. I still have to think about it. But consider the following: If $U \to X$ is part of a covering family in the site of open subsets of a space $X$, then $U$ is intuitively a part of $X$ and it makes sense to ask whether $\mathcal{F}(X) \to \mathcal{F}(U)$ is surjective. However, if instead $U \to X$ is part of a covering family in an arbitrary site, then $U$ is not necessarily a part of $X$ and the question isn’t as meaningful. For example, consider the étale covering $X \amalg X \to X$. Almost always $\mathcal{F}(X \amalg X) = \mathcal{F}(X)^2 \to \mathcal{F}(X)$ will not be surjective.
makes me wonder if this concept really makes sense in the non-localic case.
But presently the entry does not even state it in the generality of locales.
Ah, okay. I’ll fix that; very minor changes suffice for this (replace “open set” by “open” and “for every $x \in X$ there exists an open neighbourhood of $x$” by “there exists a covering of $X$”).
Added the formulation in the localic case and added the link to partial map classifier: The object $P_{\leq 1}(F)$ of subsingletons of $F$ classifies partial maps into $F$.
Is there a generalization of this notion which does not require it to be a sheaf? A quasi-topos has uniqueness, but not existence. Is there a dual notion which has existence, but not uniqueness? The context is the Charles Rezk’s answer on the connection between sSets and sheaves.
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