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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.
Could we add something about Godement resolutions here? That’s where the monad concept first makes it’s appearance, in 1958, no?
I brought this construction up a few months ago, here. Hmm, so why do we put the emphasis on comonad at Roger Godement?
@IngoBlechschmidt Here is such a reference: First Steps in Synthetic Guarded Domain Theory: Step-Indexing in the Topos of Trees
What they use flabbiness / “totality” for is the following. In this topos (presheaves on $\omega$), there is an operator $\rhd \phi$ on propositions (pronounced “later $\phi$”) which has $\phi \supset \rhd\phi$ and $(\rhd\phi\supset\phi)\supset\phi$ which is used to reason about fixed points, which are fundamental in computer science. In addition to commuting with conjunction and implication and binary disjunction, there are two useful interactions of this modality with the quantifiers, fixing $\phi:X\times Y\to\Omega$:
Now, suppose that $Y$ is flabby/”total”; then (2) holds in the opposite direction. Supposing that $Y$ is flabby/”total” and is additionally inhabited (in the sense that $\exists y:Y.\top$ is true, not in the sense of having a global element), then (1) holds in the opposite direction.
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