Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab nonassociative noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2016

    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.)

    • CommentRowNumber2.
    • CommentAuthoramg
    • CommentTimeMar 13th 2016

    I just made a few additions/modifications. The exit path \infty-category actually comes via a certain \infty-categorical localization 𝒮trat\mathcal{S}trat of the 1-category StratStrat (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”.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 13th 2016

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

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeMar 13th 2016

    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.

    • CommentRowNumber5.
    • CommentAuthoramg
    • CommentTimeMar 13th 2016
    • (edited Mar 13th 2016)

    @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 (,n)(\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 1 2\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 (,n)(\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).

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMar 13th 2016

    Thanks!

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 27th 2017
    • (edited Oct 27th 2017)

    I thought stratified homotopy hypothesis deserved a page to itself.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 30th 2017

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

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeOct 30th 2017

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

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)