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I have created stratified space in order to collect some references
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?
I added excellent notes
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.
@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).
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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