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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 $U \to X$ 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.

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeMar 28th 2019

Added recent references that discuss flabbiness and related injectivity internally.

• CommentRowNumber10.
• CommentAuthorjonsterling
• CommentTimeMar 31st 2019
In mathematics, flabbiness seems to have been considered mostly for sheaves, but I am recalling that flabbiness also comes up in _presheaf models_ that type theorists use to explain programming languages; in this area of the literature, such a presheaf is sometimes called "total". For instance, in the topos of trees, the flabby presheaves have a useful property -- a certain modality (the "later modality") can be commuted with the existential quantifier over a flabby object. Is this of interest?
5. @jonsterling: This is most definitely of interest! Can you recommend any pointers?
• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeMar 31st 2019

Added citation to Kock (thanks Ingo)

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeApr 1st 2019

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?

• CommentRowNumber14.
• CommentAuthorjonsterling
• CommentTimeApr 1st 2019
• (edited Apr 1st 2019)

@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 $\phi:X\times Y\to\Omega$:

1. $x : X \mid (\exists y:Y.\rhd \phi(x,y)) \vdash \rhd\exists y:Y.\phi(x,y)$
2. $x : X \mid \rhd(\forall y:Y.\phi(x,y))\vdash \forall y:Y.\rhd\phi(x,y))$

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.

• CommentRowNumber15.
• CommentAuthormaxsnew
• CommentTimeAug 4th 2022

Add a redirect for flabby object, and point out earlier in the page that this can be defined in any topos.

In fact should the name of the page be changed to flabby object?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeAug 6th 2022

In fact should the name of the page be changed to flabby object?

The present form of the entry wouldn’t justify renaming it, as the definition of “flabby object” is at best implicit inside nested clauses of Prop. 2.1, no? Also, it seems clearly useful to have an entry dedicated to flabby sheaves. But starting a new entry flabby object might be good.

• CommentRowNumber17.
• CommentAuthorHurkyl
• CommentTimeAug 6th 2022
• (edited Aug 6th 2022)

FWIW, there are a lot of concepts that, on the nLab, have both a “foo” and a “foo object” page.