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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 29th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have created _[[stratified space]]_ in order to collect some references

   I have created stratified space in order to collect some references

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 29th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI 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](http://golem.ph.utexas.edu/category/2009/09/what_do_mathematicians_need_to_1.html#c027263). It's [Transversal homotopy theory](http://arxiv.org/abs/0910.3322). Did anything come of that?

   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?

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeMay 29th 2012
   • (edited May 29th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 1st 2012
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex@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 $\infty Gpd$ 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 $Set$ 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).

   @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 $\infty Gpd$ 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 $Set$ 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).

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 1st 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo the Baez-Dolan approach is different. Paths cross strata, not just exit them. Woolf had already done [something](http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c020261) along the lines of Treumann.

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