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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.)
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”.
Should any of that just added be on the parent page stratified space, or is it all poset-stratified specific?
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.
@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).
Thanks!
I thought stratified homotopy hypothesis deserved a page to itself.
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 $(\infty, 1) Cat$?
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.
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