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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.
this edit adds a basic definition of stratified spaces as decompositions of spaces into strata, and then discusses how this definition links to other simple definitions (poset-stratifications, filtrations).
it also explains how to define the category of stratifications, and how this category is related to the category of posets by functors in both directions.
it adds a discussion of an “nPOV” on stratifications, explaining the analogy sets:spaces::posets:stratifications. It mentions two constructions of fundamental categories of stratifications.
some technicalities (e.g. working with convenient topological spaces for some of the definitions) have been omitted. typos and mistakes have been left to the reader to find.
have touched wording and hyperlinking in the the first sections (up to and ex-cluding “Fundamental categories”)
the section labeled “Basic features – Stratification structures” looks like it is the Definition-section. I’d suggest to name that section “Definition”, especially since a section with that title is otherwise missing
replaced “Wiki entry” by “Wikipedia entry” (NB: “wiki” is a general term, e.g. the nLab is an instance of a wiki)
replaced all references to the “next remark” etc. by anchored pointers.
the numbered “Remark” which introduces the characteristic map (here) looks like a definition to me, while the unnumbered paragraph that follows it looks like a remark.
replaced bare occurrences of “space” with “topological space”. The distinction generally matters, and an entry on stratified space is one place where it does, better to be extra clear about it.
in the “Example” labeled “Poset stratification” (here) where it says “…determines a stratification…” it seems crucial to be specific about which notion of stratification is meant now. I suppose what is meant is the previous definition, so I added an anchored pointer.
unfortunately our Instiki parser is funny about handling the symbol “|
”: It tends to mess up the spacing unless one presents the symbol as an “operator” such as in “\left| ... \right|
” (or “\big| ... \big|
” etc. if that’s what one really wants). It seems that enclosing it as “{|...|}
” also fixes the issue, but I feel that “\left| ... \right|
” is probably safer; in any case, that’s what i implemented now, throughout.
[edit: I see that the same problem exists for “\parallel
” and that in this case only the solution “{\parallel ... \parallel}
” works.]
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