Timeline for generalisations of the Seifert-van Kampen Theorem?
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13 events
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| Aug 10, 2012 at 14:21 | comment | added | Ronnie Brown | @Marc: If you read my book "Topology and Groupoids" you will see, I hope, the practical value of having a set of base points available, something I have advocated since the 1960s. In fact writing the first edition of the book which is now T&G convinced me that all to do with 1-dimensional homotopy theory was better expressed in terms of groupoids rather than groups, and so led to the question of whether groupoids could be useful in higher homotopy theory. The work with Philip Higgins showed how the proof of the 1-dimensional theorem generalised to all dimensions, giving old and new results. | |
| Jul 17, 2012 at 2:51 | comment | added | Akhil Mathew | I tried to elaborate on Mike Shulman's comment in the comment space to his answer (i.e., that it is sufficient to restrict to the $n+1$-truncation for $n$-groupoids). | |
| Jul 16, 2012 at 23:05 | comment | added | Marc Hoyois | Akhil computes the fundamental group of $S^1$ without using any set of base points. Once you know that such colimits (of full fundamental groupoids) are homotopy colimits, choosing a set of base points that intersects every open in your diagram only modifies the groupoids in the diagram (and hence their colimit) up to equivalence, so it's not a useful operation! As Mike remarks in his answer, the point of 3-fold intersections is that we can disregard 4-fold intersections and higher without changing the homotopy colimit (this much should remain true for higher $n$, replacing $3$ by $n+2$). | |
| Jul 16, 2012 at 19:25 | comment | added | Ronnie Brown | That seems the right kind of idea, and is easy for the pushout case. Question: where does the 3-fold intersection come in for the general case? This is a test! The point is that from many practical points of view one does want a set of base points, as Akhil's example for $S^1$ makes clear. (To get this example was of course my reason for studying groupoids in the 1960s.) See the result in Section 8.4 of "Topology and Groupoids" which is applied in Section 9.2 in a proof of the Jordan Curve Theorem. The next test is the 2-d SvKT for second relative homotopy groups. | |
| Jul 16, 2012 at 18:58 | comment | added | Marc Hoyois | @Ronnie: The connectivity conditions are just to guarantee that the fundamental groupoids of $U$, $V$, and $U\cap V$ are equivalent to the fundamental groups on the base point (viewed as groupoids with one object). Since homotopy pushouts preserve equivalences, the pushout of these groups is the fundamental group of $U\cup V$. Leaving out the connectivity conditions just gives a more powerful result at no additional cost. | |
| Jul 16, 2012 at 17:10 | comment | added | Ronnie Brown | When a theorem has certain conditions, a "proof" has to show precisely how those conditions are used --that is fairly basic! This is important because our HHSvKTs have essential connectivity conditions, and indeed prove them for certain colimits of structured spaces, something not at all trivial in some cases, so coping with connectivity conditions is part of the job of "generalisations" of the SvKT. There may be ways for Lurie's statement to be useful in proving our results, but that has not been shown so far. | |
| Jul 16, 2012 at 16:46 | comment | added | Ronnie Brown | The fundamental groupoid is not the same as the fundamental groupoid on a set of base points, or the fundamental group for a space with base point: I started in the 1960s with the result for the full fundamental groupoid, and then realised something more was needed to compute the fundamental group of $S^1$. Akhil's argument does not mention any connectivity conditions, and so does not, at present, or precisely, recover the classical SvKT! It is not enough to say "stated for groupoids rather than groups, though". | |
| Jul 16, 2012 at 15:47 | comment | added | Marc Hoyois | The strict pushout of the fundamental groupoids will be a homotopy pushout, because the model structure on groupoids is left proper. So Akhil's argument does recover the classical SvKT. | |
| Jul 16, 2012 at 11:07 | comment | added | Akhil Mathew | My apologies if my speculations were way off. Your work is definitely on my to-read list, but I haven't yet gotten there... | |
| Jul 16, 2012 at 9:20 | comment | added | Ronnie Brown | I would like to emphasise that my work on higher homotopy SvKTs deals with strict colimits of strict structures: the work with Philip Higgins enables (some!) complete determinations of relative homotopy groups as modules (crossed if $n=2$) over fundamental groupoids, including nonabelian second relative homotopy groups; the work with Jean-Louis Loday enables (some!) complete determination of $n$-adic homotopy groups, which are also nonabelian at the bottom level. The proofs do use (strict) higher homotopy groupoids; all may be quite distinct from the aims of "Higher Algebra". | |
| Jul 16, 2012 at 0:47 | comment | added | David Roberts♦ | The subtlety is that the work of Ronnie and collaborators deals entirely with colimits, but Lurie's A.3.1 is a homotopy colimit. Groupoids still form a 2-category, so the truncation functor would, I expect, recover the pushout as a 2-colimit. The question is more about recovering the exact statement of the classical SvKT from the version with singular simplices, in particular, how one gets the necessary assumptions for the SvKT to hold, which are tight, from the pretty much assumptionless A.3.1 | |
| Jul 15, 2012 at 23:53 | history | edited | David White | CC BY-SA 3.0 |
Minor edits
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| Jul 15, 2012 at 22:28 | history | answered | Akhil Mathew | CC BY-SA 3.0 |