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-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex 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 foundation foundations functional-analysis functor galois-theory 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 homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory 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 noncommutative noncommutative-geometry number-theory 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 sheaves simplicial space spin-geometry stable-homotopy-theory string string-theory 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.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2012

    I have created stratified space in order to collect some references

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 29th 2012

    I see that links to the fundamental category with duals of a stratified space. That old Cafe discussion led to a paper by Woolf, as John mentioned here. It’s Transversal homotopy theory.

    Did anything come of that?

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeMay 29th 2012
    • (edited May 29th 2012)

    I added excellent notes

    • M. Banagl, Topological invariants of stratified spaces, Springer Monographs in Math. 2000.

    As a graduate student in Wisconsin, I was among the guinea pigs who listened an excellent and clear exposition by the author of parts of the notes (directed toward the intersection cohomology) , before they were finalized.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 1st 2012

    @David - I’m not sure, but Lurie has some material in appendix A to Higher Algebra on what he calls the exit-path \infty-category. I think this is a generalisation of the 2-category described by Treumann in arXiv:0708.0659 and the results therein. Essentially representations of the exit path \infty-category in Gpd\infty Gpd are the same as constructible \infty-sheaves, generalising the case of representations of the fundamental \infty-groupoid being the same as locally constant \infty-sheaves. This is of course a massive generalisation of the old result that representations of the fundamental groupoid in SetSet give covering spaces.

    I should say that ’constructible’ just means ’locally constant on each stratum’. The 1-stack of perverse sheaves (a subcategory of the derived category of coherent sheaves) is an example of a constructible 1-stack.

    There is a van Kampen theorem for the exit-path \infty-category, which I like to think of as the ultimate version of Ronnie Brown’s work on van Kampen-type results on filtered spaces (which give rise to a natural stratification).

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 1st 2012

    So the Baez-Dolan approach is different. Paths cross strata, not just exit them. Woolf had already done something along the lines of Treumann.

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)