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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexCreated [[poset-stratified space]]. I wasn't sure what to call this, since the references generally just call it a "stratified space", but our page [[stratified space]] is about *all* notions of stratified space rather than just one of them. Suggestions are welcome. (Am now listening to John Francis talk about these things at the [Mid-Atlantic Topology Conference](http://www.math.jhu.edu/matc/).)

   Created poset-stratified space. I wasn’t sure what to call this, since the references generally just call it a “stratified space”, but our page stratified space is about all notions of stratified space rather than just one of them. Suggestions are welcome.

   (Am now listening to John Francis talk about these things at the Mid-Atlantic Topology Conference.)

2. • CommentRowNumber2.
   • CommentAuthoramg
   • CommentTimeMar 13th 2016
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   Author: amg
   Format: MarkdownItexI just made a few additions/modifications. The exit path $\infty$-category actually comes via a certain $\infty$-categorical localization $\mathcal{S}trat$ of the 1-category $Strat$ (which is itself actually now _conically smooth_ stratified spaces, not all stratified spaces). So I added just enough to correctly state the result, and I also added the statement of the "stratified homotopy hypothesis".

   I just made a few additions/modifications. The exit path $\infty$-category actually comes via a certain $\infty$-categorical localization $\mathcal{S}trat$ of the 1-category $Strat$ (which is itself actually now conically smooth stratified spaces, not all stratified spaces). So I added just enough to correctly state the result, and I also added the statement of the “stratified homotopy hypothesis”.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 13th 2016
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   Author: David_Corfield
   Format: MarkdownItexShould any of that just added be on the parent page [[stratified space]], or is it all poset-stratified specific?

   Should any of that just added be on the parent page stratified space, or is it all poset-stratified specific?

4. • CommentRowNumber4.
   • CommentAuthorTim_Porter
   • CommentTimeMar 13th 2016
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   Author: Tim_Porter
   Format: MarkdownItexHow does this stuff relate to [[proper homotopy theory]]? There are various versions of that but Baues had one in which each space had an infinite tree that played the role of the base point, and he considered spherical objects of various types which formed a (Lawvere) theory whose algebras (I think) were the analogues of groups in this context.

   How does this stuff relate to proper homotopy theory? There are various versions of that but Baues had one in which each space had an infinite tree that played the role of the base point, and he considered spherical objects of various types which formed a (Lawvere) theory whose algebras (I think) were the analogues of groups in this context.

5. • CommentRowNumber5.
   • CommentAuthoramg
   • CommentTimeMar 13th 2016
   • (edited Mar 13th 2016)
   • PermaLink
   Author: amg
   Format: MarkdownItex@David: This all has to do with stratified spaces as developed by Ayala--Francis--Tanaka and furthered by Ayala--Francis--Rozenblyum. Their primary motivations are rooted in the theory of $(\infty,n)$-categories, for which stratified spaces are a key technical tool. So it's definitely all specific to the definitions given at the new page that Mike created, though if I recall correctly these are compared with other definitions (notably Whitney's) in one of the Ayala--Francis--Tanaka papers. I think their definition is less general, so as to eliminate certain pathological phenomena (I'm recalling a non-proper embedding $\mathbb{R}^1 \hookrightarrow \mathbb{R}^2$ that spirals in towards the origin). @Tim: I could easily be missing something, but I don't immediately see a connection. Again, the motivation isn't really to study stratified spaces themselves, but to use them as a link between smooth manifolds and $(\infty,n)$-categories. By the way, prompted by David's link to the main "stratified spaces" page, I made a small addition there to reference this "exit path $\infty$-category construction" (though I wasn't immediately able to figure out how to link to a section within a page, so I just faked it).

   @David: This all has to do with stratified spaces as developed by Ayala–Francis–Tanaka and furthered by Ayala–Francis–Rozenblyum. Their primary motivations are rooted in the theory of $(\infty,n)$-categories, for which stratified spaces are a key technical tool. So it’s definitely all specific to the definitions given at the new page that Mike created, though if I recall correctly these are compared with other definitions (notably Whitney’s) in one of the Ayala–Francis–Tanaka papers. I think their definition is less general, so as to eliminate certain pathological phenomena (I’m recalling a non-proper embedding $\mathbb{R}^1 \hookrightarrow \mathbb{R}^2$ that spirals in towards the origin).

   @Tim: I could easily be missing something, but I don’t immediately see a connection. Again, the motivation isn’t really to study stratified spaces themselves, but to use them as a link between smooth manifolds and $(\infty,n)$-categories.

   By the way, prompted by David’s link to the main “stratified spaces” page, I made a small addition there to reference this “exit path $\infty$-category construction” (though I wasn’t immediately able to figure out how to link to a section within a page, so I just faked it).

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMar 13th 2016
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   Author: Mike Shulman
   Format: MarkdownItexThanks!

   Thanks!

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 27th 2017
   • (edited Oct 27th 2017)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI thought [[stratified homotopy hypothesis]] deserved a page to itself.

   I thought stratified homotopy hypothesis deserved a page to itself.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 30th 2017
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   Author: David_Corfield
   Format: MarkdownItexApart from doing the honest toil, is there anything likely to stand in the way of cohesive $(\infty, 2)$-toposes, defined via an adjoint quadruple to $(\infty, 1) Cat$?

   Apart from doing the honest toil, is there anything likely to stand in the way of cohesive $(\infty, 2)$-toposes, defined via an adjoint quadruple to $(\infty, 1) Cat$?

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeOct 30th 2017
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   Author: Mike Shulman
   Format: MarkdownItexI don't think so: e.g. adjoint product-preserving functors between cartesian monoidal $(\infty,1)$-categories should induce adjoint functors between the corresponding $(\infty,2)$-categories of enriched $(\infty,1)$-categories.

   I don’t think so: e.g. adjoint product-preserving functors between cartesian monoidal $(\infty,1)$-categories should induce adjoint functors between the corresponding $(\infty,2)$-categories of enriched $(\infty,1)$-categories.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 13th 2023
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    Author: David_Corfield
    Format: MarkdownItexAdded a reference (seems to be about the poset-stratified case): * [[Peter J. Haine]], _On the homotopy theory of stratified spaces_ ([arXiv:1811.01119](https://arxiv.org/abs/1811.01119)) <a href="https://ncatlab.org/nlab/revision/diff/poset-stratified+space/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/poset-stratified+space/9">v9</a>, <a href="https://ncatlab.org/nlab/show/poset-stratified+space">current</a>

    Added a reference (seems to be about the poset-stratified case):

    • Peter J. Haine, On the homotopy theory of stratified spaces (arXiv:1811.01119)

    diff, v9, current