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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

1. Added to flabby sheaf several characterizations of flabbiness, an external one which, unlike the usual definition, is manifestly local, and several internal ones.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 29th 2016

Thanks!

(Should the entry not state the definition in more generality than over sites of opens?)

• CommentRowNumber3.
• CommentAuthorZhen Lin
• CommentTimeMar 29th 2016

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.

2. 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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 29th 2016

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.

3. 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 $X$”).

4. 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 $F$.

• CommentRowNumber8.
• CommentAuthorspitters
• CommentTimeNov 8th 2016

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.