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  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 UXU \to X is part of a covering family in the site of open subsets of a space XX, then UU is intuitively a part of XX and it makes sense to ask whether (X)(U)\mathcal{F}(X) \to \mathcal{F}(U) is surjective. However, if instead UXU \to X is part of a covering family in an arbitrary site, then UU is not necessarily a part of XX and the question isn’t as meaningful. For example, consider the étale covering X⨿XXX \amalg X \to X. Almost always (X⨿X)=(X) 2(X)\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 xXx \in X there exists an open neighbourhood of xx” by “there exists a covering of XX”).

  4. Added the formulation in the localic case and added the link to partial map classifier: The object P 1(F)P_{\leq 1}(F) of subsingletons of FF classifies partial maps into FF.

    • 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.

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