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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeNov 22nd 2014
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   Author: Mike Shulman
   Format: MarkdownItexSuppose we have a cohesive $(\infty,1)$-topos $\mathbf{H}$. We've established that at least in most examples, the shape modality $ʃ$ can be extended to a stable factorization system, i.e. to an internal modality, which therefore supplies a reflective subcategory of "discrete objects" in each slice $\mathbf{H}/X$. However, we don't know how to extend $\flat$ to an operation of any sort on slices, let alone one that coreflects into this subcategory. Here's a simpler question: if $Y$ is an object of $\mathbf{H}$, we have the coreflection $\flat Y \to Y$, and we can pull it back to $\mathbf{H}/X$. Is the resulting map $X\times \flat Y \to X\times Y$ a coreflection of $X\times Y$ into the discrete objects of $\mathbf{H}/X$?

   Suppose we have a cohesive $(\infty,1)$-topos $\mathbf{H}$. We’ve established that at least in most examples, the shape modality $ʃ$ can be extended to a stable factorization system, i.e. to an internal modality, which therefore supplies a reflective subcategory of “discrete objects” in each slice $\mathbf{H}/X$. However, we don’t know how to extend $\flat$ to an operation of any sort on slices, let alone one that coreflects into this subcategory.

   Here’s a simpler question: if $Y$ is an object of $\mathbf{H}$, we have the coreflection $\flat Y \to Y$, and we can pull it back to $\mathbf{H}/X$. Is the resulting map $X\times \flat Y \to X\times Y$ a coreflection of $X\times Y$ into the discrete objects of $\mathbf{H}/X$?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 23rd 2014
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   Author: Mike Shulman
   Format: MarkdownItexThe answer is no. In fact, I now have a proof (checked in Coq) that fiberwise coreflections _cannot_ exist, and it only needs coreflections of this sort. More precisely, any "coreflective subuniverse" in HoTT is of the form $(-\times U)$ for some hprop $U$. In particular, if it preserves $1$ (as it would have to, if it coreflected into the discrete objects), then it is the identity.

   The answer is no.

   In fact, I now have a proof (checked in Coq) that fiberwise coreflections cannot exist, and it only needs coreflections of this sort. More precisely, any “coreflective subuniverse” in HoTT is of the form $(-\times U)$ for some hprop $U$. In particular, if it preserves $1$ (as it would have to, if it coreflected into the discrete objects), then it is the identity.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 23rd 2014
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   Author: David_Corfield
   Format: MarkdownItexIs there a 'moral' to this story? Perhaps one where we come to see the inevitability of why we can't have what we wanted, but realise that there's a better path to take anyway. Something like the one of overcoming the disappointment of not being able to reach a contradiction from denying the parallel postulate by realising that this shows a richer array of geometries.

   Is there a ’moral’ to this story? Perhaps one where we come to see the inevitability of why we can’t have what we wanted, but realise that there’s a better path to take anyway. Something like the one of overcoming the disappointment of not being able to reach a contradiction from denying the parallel postulate by realising that this shows a richer array of geometries.

4. • CommentRowNumber4.
   • CommentAuthorTim_Porter
   • CommentTimeNov 23rd 2014
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   Author: Tim_Porter
   Format: MarkdownItexIs this `categorical pragmatism'?

   Is this ‘categorical pragmatism’?

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 23rd 2014
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   Author: David_Corfield
   Format: MarkdownItexThat sounds close to an oxymoron, at least if we're following in the path of Grothendieck (and you mean pragmatism in the popular sense).

   That sounds close to an oxymoron, at least if we’re following in the path of Grothendieck (and you mean pragmatism in the popular sense).

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeNov 23rd 2014
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   Author: Mike Shulman
   Format: MarkdownItexI wish there were a moral like that. But if there is, I don't think I've seen it yet.

   I wish there were a moral like that. But if there is, I don’t think I’ve seen it yet.