Timeline for generalisations of the Seifert-van Kampen Theorem?
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12 events
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| Jan 10, 2013 at 11:05 | comment | added | Ronnie Brown | Maybe there is confusion about the purposes of the Seifert-van Kampen Theorems. One aim is to compute homotopy invariants, which show that certain spaces or structured spaces are not of the same homotopy type. For this it is important to be able to make explicit computations. That is one reason for working with strict structures. The literature on the nonabelian tensor product of groups (126 items listed on pages.bangor.ac.uk/~mas010/nonabtens.html ) gives many examples of such calculations. | |
| Jul 18, 2012 at 15:55 | comment | added | Ronnie Brown | The great thing about the limited algebraic model of crossed complexes (or its various multi-groupoid equivalent structures) is not only the SvKT but also the monoidal closed structure, related to homotopies, and products. The (strict) cat$^n$-structures are more powerful, do model homotopy $(n+1)$-types, and are related via crossed $n$-cubes of groups to classical ideas of $r$-ad homotopy groups, $r \leqslant n$ in rather a marvellous way. One difficulty is to get at homotopies, or monoidal structures. I would like to see work on bifiltered spaces as an experiment. | |
| Jul 17, 2012 at 17:38 | comment | added | Mike Shulman | Finally, I agree, I would very much like to know whether higher-dimensional strict versions can be extracted from Lurie's theorem. The first steps of the proof I gave should work just as well for arbitrary $n>1$, but it seems that the identification of weak and strict colimits of $n$-groupoids may fail because we don't have a good model structure on algebraic $n$-groupoids, and if we did then its cofibrations would probably be hard to understand. Maybe your cubical approaches could be rephrased as a solution to this problem? | |
| Jul 17, 2012 at 17:34 | comment | added | Mike Shulman | Also, personally, when I say or hear "A generalizes B", I generally assume that this means in particular that "A implies B". In this sense, Lurie's theorem is not strictly speaking a generalization of the most general possible SvKT with a set of basepoints (at least, I don't see how). However, in the text surrounding the theorem, he only claims that it generalizes the most classical SvKT about unions of two open subsets, and this does seem to be true. | |
| Jul 17, 2012 at 17:30 | comment | added | Mike Shulman | I certainly don't mean to present this as the best way to prove the SvKT! By no means should a beginner in algebraic topology be expected to know about (2,1)-toposes, Reedy model categories, or even transfinite induction. On the other hand, too much emphasis on concrete and model-dependent statements and proofs tends to obscure the acquiring of a high-level view of a subject, so I think a balance needs to be struck (and the right balance will always come down to a matter of personal preference). | |
| Jul 17, 2012 at 16:16 | comment | added | Ronnie Brown | I should recall that in the first 1968 edition of my book now "Topology and Groupoids", the argument that a retract of a pushout is a pushout was used there to compute $\pi_1(S^1)$ perhaps prompting the then MAA reviewer to write: "This reads like a book on topology written by a category theorist." Another local-to-global result in the current edition, and as tricky as the SvKT, computes $\pi_1(X/G)$ as $(\pi_1X)//G$ (orbit groupoid) for certain useful situations. Is there a generalisation of this result to higher dimensions? or other generalisations? | |
| Jul 17, 2012 at 10:56 | comment | added | Ronnie Brown | Mike's seems a good complete answer to my question, though I think beginners in algebraic topology might prefer the proof of the general result in our paper Archiv. Math. 42 (1984) 85-88! Part of my confusion behind my question is the distinction between "A generalises B", and "A implies B", especially if A also implies excision, MV, .... Mike's proof leaves open the question of proofs of the 2-d and higher dimensional results: the current proofs involve new methods, but the theorems directly imply classical results in algebraic topology, e.g. on $\pi_n(S^n)$, and on free crossed modules. | |
| Jul 17, 2012 at 2:51 | comment | added | Akhil Mathew | (Er, I meant $n+2$ and $n+1$-category.) | |
| Jul 17, 2012 at 2:48 | comment | added | Akhil Mathew | ...intersections of the $\{U_\alpha\}$, and as you pointed out this can be calculated by restricting to the $n+1$-skeleton (since we are in an $n$-category). At least, I think this elaborates on your first comment----I'm going to need to digest this answer now. | |
| Jul 17, 2012 at 2:47 | comment | added | Akhil Mathew | To add to this answer: let $\{U_\alpha\}_{\alpha \in A}$ be an open covering of a topological space $X$. Then we can form a simplicial space which in degree zero is $\bigsqcup_{\alpha \in A} U_\alpha$, which in degree one is $\bigsqcup_{\alpha \neq \beta} U_\alpha \cap U_\beta$, and so forth: the $n$-simplices come from the $n$-fold intersections. The space $X$ can (if I'm not mistaken) be recovered as the (homotopy) colimit of this whole simplicial construction. This means that the fundamental $n$-groupoid should be the homotopy colimit of the fundamental $n$-groupoids of various... | |
| Jul 16, 2012 at 23:06 | comment | added | David Roberts♦ | Bravo! Nice to see this sorted out. | |
| Jul 16, 2012 at 22:04 | history | answered | Mike Shulman | CC BY-SA 3.0 |