Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMar 26th 2016
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexAdded to _[[flabby sheaf]]_ several characterizations of flabbiness, an external one which, unlike the usual definition, is manifestly local, and several internal ones.

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 29th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! (Should the entry not state the definition in more generality than over sites of opens?)

   Thanks!

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

3. • CommentRowNumber3.
   • CommentAuthorZhen Lin
   • CommentTimeMar 29th 2016
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexIt'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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMar 29th 2016
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexRight. 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 29th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMar 30th 2016
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexAh, 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 $X$").

   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 $X$”).

7. • CommentRowNumber7.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeApr 4th 2016
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexAdded 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 $F$.

   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 $F$.

8. • CommentRowNumber8.
   • CommentAuthorspitters
   • CommentTimeNov 8th 2016
   • PermaLink
   Author: spitters
   Format: MarkdownItexIs 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](http://mathoverflow.net/questions/5236/abstract-relation-between-presehaves-and-simplicial-sets).

   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.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMar 28th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAdded recent references that discuss flabbiness and related injectivity internally. <a href="https://ncatlab.org/nlab/revision/diff/flabby+sheaf/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/flabby+sheaf/16">v16</a>, <a href="https://ncatlab.org/nlab/show/flabby+sheaf">current</a>

   Added recent references that discuss flabbiness and related injectivity internally.

   diff, v16, current

10. • CommentRowNumber10.
    • CommentAuthorjonsterling
    • CommentTimeMar 31st 2019
    • PermaLink
    Author: jonsterling
    Format: TextIn 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?

    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?
11. • CommentRowNumber11.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeMar 31st 2019
    • PermaLink
    Author: IngoBlechschmidt
    Format: Text@jonsterling: This is most definitely of interest! Can you recommend any pointers?

    @jonsterling: This is most definitely of interest! Can you recommend any pointers?
12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeMar 31st 2019
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAdded citation to Kock (thanks Ingo) <a href="https://ncatlab.org/nlab/revision/diff/flabby+sheaf/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/flabby+sheaf/17">v17</a>, <a href="https://ncatlab.org/nlab/show/flabby+sheaf">current</a>

    Added citation to Kock (thanks Ingo)

    diff, v17, current

13. • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 1st 2019
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexCould we add something about [Godement resolutions](https://en.wikipedia.org/wiki/Godement_resolution) 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](https://nforum.ncatlab.org/discussion/9302/modalities-induced-by-cohesive-counits/). Hmm, so why do we put the emphasis on *co*monad at [[Roger Godement]]?

    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?

14. • CommentRowNumber14.
    • CommentAuthorjonsterling
    • CommentTimeApr 1st 2019
    • (edited Apr 1st 2019)
    • PermaLink
    Author: jonsterling
    Format: MarkdownItex@IngoBlechschmidt Here is such a reference: [First Steps in Synthetic Guarded Domain Theory: Step-Indexing in the Topos of Trees](https://arxiv.org/pdf/1208.3596.pdf) 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.

    @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.

15. • CommentRowNumber15.
    • CommentAuthormaxsnew
    • CommentTimeAug 4th 2022
    • PermaLink
    Author: maxsnew
    Format: MarkdownItexAdd 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? <a href="https://ncatlab.org/nlab/revision/diff/flabby+sheaf/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/flabby+sheaf/20">v20</a>, <a href="https://ncatlab.org/nlab/show/flabby+sheaf">current</a>

    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?

    diff, v20, current

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItex> 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.

    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.

17. • CommentRowNumber17.
    • CommentAuthorHurkyl
    • CommentTimeAug 6th 2022
    • (edited Aug 6th 2022)
    • PermaLink
    Author: Hurkyl
    Format: MarkdownItexFWIW, there are a lot of concepts that, on the nLab, have both a "foo" and a "foo object" page.

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