Timeline for Exact sequences in homotopy categories
Current License: CC BY-SA 3.0
5 events
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| Mar 1, 2012 at 11:09 | comment | added | Ronnie Brown | We got going on higher SvKTs when we realised a 2-d version could profitably involve a double groupoid functor from pointed (or many-pointed) pairs of spaces to double groupoids, so yielding a SvKT for second relative homotopy groups as crossed modules, moving the classical proof up one dimension. So far, nobody has been able to deduce this theorem from some "$\tau$ is a left adjoint" principle! This result is also related to excision, but involving actions. The main point was getting hold of functors which could model algebraically the geometry of the proof! The rest followed. | |
| Feb 29, 2012 at 22:58 | comment | added | Marc Hoyois | You're right, I should have said "maybe". I have absolutely zero knowledge of $\geq 2$ van Kampen theorems, so that was just my mathematical optimism speaking. Certainly the fact that $\tau_{\leq n}$ preserves homotopy colimits is a natural generalization of the ususal van Kampen from $1$ to $n$, but I don't know to what extent it matches so-called higher van Kampen theorems. My uneducated guess is that it is the "homotopical essence" of higher SvK theorems, and that the rest is getting your hands on $n$-groupoids. | |
| Feb 29, 2012 at 21:01 | comment | added | Ronnie Brown | I find the word "probably" just a bit airy-fairy, or maybe just a statement of a research project! Start with the 2-dimensional SvKT, which computes precisely some nonabelian second relative homotopy groups as crossed modules. Examples are in (RB and C.D.WENSLEY), `Computation and homotopical applications of induced crossed modules', J. Symbolic Computation 35 (2003) 59-72, also in the book "Nonabelian algebraic topology". And many precise computations for 3-types are in the literature on the nonabelian tensor product of groups, using the algebraic structures underlying the geometry. | |
| Feb 29, 2012 at 12:47 | comment | added | Marc Hoyois | I think the result I referenced from HTT in my answer is a strict (and vast) generalization of SvK (and probably of its higher-n versions, modulo the computation of homotopy colimits of $n$-groupoids). A pushout of groupoids along inclusions is a homotopy pushout, so you directly recover the usual SvK, but it also works for open covers with more patches, quotients by free group actions, etc. As to your last sentence, I find the proof in HTT pretty abstract :) | |
| Feb 29, 2012 at 9:38 | history | answered | Ronnie Brown | CC BY-SA 3.0 |