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    • 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.

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