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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2012
   • (edited Apr 16th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex[just a moment, re-editing message...]

   [just a moment, re-editing message…]

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2012
   • (edited Apr 16th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSecond attempt, my first message contained a stupidity. And this one here will not be very deep either. It's just that when running through the park tonight, I started wondering about the following: can we somehow realize Segal space objects in an $\infty$-topos $\mathbf{H}$ -- or something close to them -- as models of a geometric theory, hence as something classified by geometric morphisms into another $\infty$-topos? For instance, when working out what a geometric morphism $\mathbf{H} \to Sh_\infty(\Delta^{op})$ is, one finds that flatness of the corresponding functor $X : \Delta^{op} \to \mathbf{H}$ in particular means that the colimits $$ [n] = [1] \coprod_{[0]} \cdots \coprod_{[0]} [1] $$ in $\Delta$ have to be taken to $\infty$-limits. This are precisely the Segal conditions on the simplicial object $X$. Unfortunately, $X$ being flat means more than this. It forces that Segal object to be not just the nerve of an $\infty$-category, but of a "linear posetal $\infty$-category", or something like this. But, so: can we maybe modify the classifying topos $Sh_\infty(\Delta^{op})$ a bit such as to actually classify general Segal objects?

   Second attempt, my first message contained a stupidity. And this one here will not be very deep either. It’s just that when running through the park tonight, I started wondering about the following:

   can we somehow realize Segal space objects in an $\infty$-topos $\mathbf{H}$ – or something close to them – as models of a geometric theory, hence as something classified by geometric morphisms into another $\infty$-topos?

   For instance, when working out what a geometric morphism $\mathbf{H} \to Sh_\infty(\Delta^{op})$ is, one finds that flatness of the corresponding functor $X : \Delta^{op} \to \mathbf{H}$ in particular means that the colimits

   $$[n] = [1] \coprod_{[0]} \cdots \coprod_{[0]} [1]$$

   in $\Delta$ have to be taken to $\infty$-limits. This are precisely the Segal conditions on the simplicial object $X$.

   Unfortunately, $X$ being flat means more than this. It forces that Segal object to be not just the nerve of an $\infty$-category, but of a “linear posetal $\infty$-category”, or something like this.

   But, so: can we maybe modify the classifying topos $Sh_\infty(\Delta^{op})$ a bit such as to actually classify general Segal objects?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2012
   • (edited Apr 16th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> can we maybe modify the classifying topos $Sh(\Delta^{op})$ a bit such as to actually classify general Segal objects? For instance, what if we take $Sh(sSet_{fin}^{op})$ instead? Then a flat functor $ X : sSet_{fin}^op \to \mathbf{H}$ will still send the colimits $\Delta[n] = \Delta[1] \coprod_{\Delta[0]} \cdots \coprod_{\Delta[0]} \Delta[1]$ to limits, hence enforce the Segal conditions. But now $\partial_0, \partial_1 : \Delta[0] \to \Delta[1]$ are no longer jointly epimorphic, hence $(d_0, d_1) : X_1 \to X_0 \times X_0$ is no longer required to be a mono. Hm...

   can we maybe modify the classifying topos $Sh(\Delta^{op})$ a bit such as to actually classify general Segal objects?

   For instance, what if we take $Sh(sSet_{fin}^{op})$ instead?

   Then a flat functor $X : sSet_{fin}^op \to \mathbf{H}$ will still send the colimits $\Delta[n] = \Delta[1] \coprod_{\Delta[0]} \cdots \coprod_{\Delta[0]} \Delta[1]$ to limits, hence enforce the Segal conditions. But now $\partial_0, \partial_1 : \Delta[0] \to \Delta[1]$ are no longer jointly epimorphic, hence $(d_0, d_1) : X_1 \to X_0 \times X_0$ is no longer required to be a mono.

   Hm…

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeApr 16th 2012
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   Author: Mike Shulman
   Format: MarkdownItexYou want the free $(\infty,1)$-category with finite limits generated by a Segal-space object. The 1-categorical analogue is well-known to exist (it's the [[syntactic category]] of the finite-limit theory of categories), but I don't think it has a clever description like you are looking for. The $(\infty,1)$-version might conjecturally be the syntactic category of a homotopy-type-theory of Segal spaces.

   You want the free $(\infty,1)$-category with finite limits generated by a Segal-space object. The 1-categorical analogue is well-known to exist (it’s the syntactic category of the finite-limit theory of categories), but I don’t think it has a clever description like you are looking for. The $(\infty,1)$-version might conjecturally be the syntactic category of a homotopy-type-theory of Segal spaces.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2012
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   Author: Urs
   Format: MarkdownItexAll right, we know abstractly that the classifying $\infty$-topos exists. It would be nice to get one's hands on it more explicitly. I was beginning to hope that it is something simple, fundamental. But maybe not.

   All right, we know abstractly that the classifying $\infty$-topos exists. It would be nice to get one’s hands on it more explicitly.

   I was beginning to hope that it is something simple, fundamental. But maybe not.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeApr 17th 2012
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   Author: Mike Shulman
   Format: MarkdownItexI think it's fairly rare to find a topos that both has a direct geometric description and classifies a familiar theory.

   I think it’s fairly rare to find a topos that both has a direct geometric description and classifies a familiar theory.

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 17th 2012
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   Author: David_Corfield
   Format: MarkdownItexRe #6, does this even ever so slightly resemble the fact that Eilenberg-Mac Lane spaces are generally geometrically complicated?

   Re #6, does this even ever so slightly resemble the fact that Eilenberg-Mac Lane spaces are generally geometrically complicated?

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeApr 17th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIt does seem similar, doesn't it? (Plus the dual fact that simple geometric spaces like spheres usually represent very complicated cohomology theories.) It seems like there's some sort of "conservation of simplicity" that simple objects often represent complicated functors, while simple functors are represented by complicated objects. Of course in that generality there are plenty of objects that are both simple and represent simple functors (and, undoubtedly, complicated objects that represent complicated functors!), but it's an interesting parallel.

   It does seem similar, doesn’t it? (Plus the dual fact that simple geometric spaces like spheres usually represent very complicated cohomology theories.) It seems like there’s some sort of “conservation of simplicity” that simple objects often represent complicated functors, while simple functors are represented by complicated objects. Of course in that generality there are plenty of objects that are both simple and represent simple functors (and, undoubtedly, complicated objects that represent complicated functors!), but it’s an interesting parallel.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2012
   • (edited Apr 17th 2012)
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   Author: Urs
   Format: MarkdownItexNow I feel like re-amplifying that I wasn't just guessing a simple classifying topos for the heck of it. Instead I thought it is kind of remarkable that the proof of the standard statement "cosimplicial sets classify inhabited linear orders" proceeds in two stages, the first being that one observes that they classify Segal objects with extra properties, the second being that these extra properties force the resulting category objects to be $(0,1)$-category objects. It seems nice and suggestive that the Segal conditions come out for free from the flatness of a functor on $\Delta^{op}$, and talking about "linear orders" here hides the fact that what is going on is not so much order-theoretic as category-theoretic, collapsing only in the last step to (0,1)-category theory. Now of course you may well be right that there is no simple way to modify this to make _only_ the Segal conditions come out, and not a stronger condition (though I'd be happy to get the Segal condition plus the completeness condition ;-). But nevertheless it seems to be something worth observing and worth thinking about for a sec.

   Now I feel like re-amplifying that I wasn’t just guessing a simple classifying topos for the heck of it.

   Instead I thought it is kind of remarkable that the proof of the standard statement “cosimplicial sets classify inhabited linear orders” proceeds in two stages, the first being that one observes that they classify Segal objects with extra properties, the second being that these extra properties force the resulting category objects to be $(0,1)$-category objects.

   It seems nice and suggestive that the Segal conditions come out for free from the flatness of a functor on $\Delta^{op}$, and talking about “linear orders” here hides the fact that what is going on is not so much order-theoretic as category-theoretic, collapsing only in the last step to (0,1)-category theory.

   Now of course you may well be right that there is no simple way to modify this to make only the Segal conditions come out, and not a stronger condition (though I’d be happy to get the Segal condition plus the completeness condition ;-). But nevertheless it seems to be something worth observing and worth thinking about for a sec.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItexAlso I think Eilenberg-MacLane objects have a very simple explicit description, if not viewed from a bad angle. I'd be quite happy if I had a description of the classifying topos for Segal objects as simple as the usual description of EM objects.

    Also I think Eilenberg-MacLane objects have a very simple explicit description, if not viewed from a bad angle. I’d be quite happy if I had a description of the classifying topos for Segal objects as simple as the usual description of EM objects.