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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 5th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexI'm not sure this question has been answered (either here or in the nLab). Famously, Urs shows that the [[Cahier topos]] is a cohesive topos; as far as I can tell, this topos is not covered in Moerdijk-Reyes (which is the only text on SDG I have in my home library). The material in the nLab on various models for SDG that are covered in that text, which go by names like $\mathcal{F}$, $\mathcal{G}$ ("the Dubuc topos"), $\mathcal{Z}$, and $\mathcal{B}$, is [pretty thin](http://ncatlab.org/nlab/show/Models+for+Smooth+Infinitesimal+Analysis). Do we know which of these models is a cohesive topos?

   I’m not sure this question has been answered (either here or in the nLab). Famously, Urs shows that the Cahier topos is a cohesive topos; as far as I can tell, this topos is not covered in Moerdijk-Reyes (which is the only text on SDG I have in my home library). The material in the nLab on various models for SDG that are covered in that text, which go by names like $\mathcal{F}$, $\mathcal{G}$ (“the Dubuc topos”), $\mathcal{Z}$, and $\mathcal{B}$, is pretty thin.

   Do we know which of these models is a cohesive topos?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 5th 2012
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   Author: Mike Shulman
   Format: MarkdownItexGood question! (-:

   Good question! (-:

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 5th 2012
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   Author: Urs
   Format: MarkdownItexThis has the same kind of answer as the analogous question about cohesive models in algebraic geometry that we are discussing every now and then in different threads: in order to get a locally connected 1-topos / locally $\infty$-connected $\infty$-topos, we need a site of definition that consists of objects which are contractible as seen by its Grothendieck topology (as discussed at [[infinity-cohesive site]]). So I think neither of these other toposes can be cohesive. But one may want to look into restricting to subsites of $C^\infty$-rings that are larger than the standard site of definition of the Cahiers topos, but still consist of only étale-contractible loci. What is currently missing is a good practical motivation to invest energy into such a search. The Cahiers topos seems to be great in practice. (Although recently I came to think that there is another sensible topology on it that I should consider.)

   This has the same kind of answer as the analogous question about cohesive models in algebraic geometry that we are discussing every now and then in different threads:

   in order to get a locally connected 1-topos / locally $\infty$-connected $\infty$-topos, we need a site of definition that consists of objects which are contractible as seen by its Grothendieck topology (as discussed at infinity-cohesive site).

   So I think neither of these other toposes can be cohesive. But one may want to look into restricting to subsites of $C^\infty$-rings that are larger than the standard site of definition of the Cahiers topos, but still consist of only étale-contractible loci.

   What is currently missing is a good practical motivation to invest energy into such a search. The Cahiers topos seems to be great in practice.

   (Although recently I came to think that there is another sensible topology on it that I should consider.)

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 5th 2012
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   Author: Mike Shulman
   Format: MarkdownItexIs it obvious that the other sites are not locally contractible? I haven't been following the algebraic geometry discussions, but my intuition is that algebraic-geometry objects are much more topologically "rigid" than smooth ones.

   Is it obvious that the other sites are not locally contractible? I haven’t been following the algebraic geometry discussions, but my intuition is that algebraic-geometry objects are much more topologically “rigid” than smooth ones.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 5th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexThanks, Urs. This is helpful to hear. Now that you mention the other threads, that does ring a bell. > we need a site of definition that consists of objects which are contractible Is it in fact a logically necessary condition? > What is currently missing is a good practical motivation to invest energy into such a search. Well, my question just sort of occurred to me so I thought I'd ask. But let's agree that what motivates people can be very subjective and personal. Obviously you shouldn't bother about it yourself if you're not motivated!

   Thanks, Urs. This is helpful to hear. Now that you mention the other threads, that does ring a bell.

   we need a site of definition that consists of objects which are contractible

   Is it in fact a logically necessary condition?

   What is currently missing is a good practical motivation to invest energy into such a search.

   Well, my question just sort of occurred to me so I thought I’d ask. But let’s agree that what motivates people can be very subjective and personal. Obviously you shouldn’t bother about it yourself if you’re not motivated!

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeOct 5th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexMy understanding is that it's not logically necessary, but that it's currently the only way we know of to produce cohesive $(\infty,1)$-toposes.

   My understanding is that it’s not logically necessary, but that it’s currently the only way we know of to produce cohesive $(\infty,1)$-toposes.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeOct 6th 2012
   • (edited Oct 6th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexRight, there is no theorem yet that a site of locally contractibles is strictly necessary. But let's look at what we know to see how much wiggle room there is: 1. The argument that in a locally connected topos every object is the coproduct of connected ones generalizes to locally $\infty$-connected toposes as follows: every object $X$ is an $\infty$-colimit of objects whose induced map into $\Pi(X)$ is null-homotopic. So at least rouhgly this says that "Every object is covered by contractibles". I used to bang my head against this triying to deduce something more explicit, but still haven't. 1. How different can $\Pi$ be from $\underset{\to}{\lim}$? The way that the existing models work is that $Const : \infty Grpd \to PSh_\infty(C)$ does factor (as $Disc$) through $Sh_\infty(C) \hookrightarrow PSh_\infty(C)$ so that then the left adjoint $\underset{\to}{\lim}$ of $Const : \infty Grpd \to PSh_\infty(C)$ restricts to the left adjoint $\Pi$ for $Disc$. But for a constant $\infty$-presheaf to satisfy descent over a representable, it must be true that first forming the Cech nerve of a covering family and then further refining that by a simplicial object which is degreewise a coproduct of representables has the property that it becomes a contractible simplicial set once all these representables are identitfied with points. So it means that every object of the site must be étale-contractible. As I said, this is assuming that $\Pi = \underset{\to}{\lim}$. Are there other possibilities for $\Pi$? We need that $Disc = Sheafify|_{Const} \circ Const$ has a left adjoint. So $Sheafify|_{Const}$ needs a left adjoint.

   Right, there is no theorem yet that a site of locally contractibles is strictly necessary. But let’s look at what we know to see how much wiggle room there is:

   1. The argument that in a locally connected topos every object is the coproduct of connected ones generalizes to locally $\infty$-connected toposes as follows: every object $X$ is an $\infty$-colimit of objects whose induced map into $\Pi(X)$ is null-homotopic. So at least rouhgly this says that “Every object is covered by contractibles”. I used to bang my head against this triying to deduce something more explicit, but still haven’t.

   2. How different can $\Pi$ be from $\underset{\to}{\lim}$?

      The way that the existing models work is that $Const : \infty Grpd \to PSh_\infty(C)$ does factor (as $Disc$) through $Sh_\infty(C) \hookrightarrow PSh_\infty(C)$ so that then the left adjoint $\underset{\to}{\lim}$ of $Const : \infty Grpd \to PSh_\infty(C)$ restricts to the left adjoint $\Pi$ for $Disc$.

      But for a constant $\infty$-presheaf to satisfy descent over a representable, it must be true that first forming the Cech nerve of a covering family and then further refining that by a simplicial object which is degreewise a coproduct of representables has the property that it becomes a contractible simplicial set once all these representables are identitfied with points. So it means that every object of the site must be étale-contractible.

      As I said, this is assuming that $\Pi = \underset{\to}{\lim}$. Are there other possibilities for $\Pi$? We need that $Disc = Sheafify|_{Const} \circ Const$ has a left adjoint. So $Sheafify|_{Const}$ needs a left adjoint.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 6th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThank you very much, Urs -- very helpful analysis.

   Thank you very much, Urs – very helpful analysis.