Timeline for Higher refinement of Seifert-van Kampen theorem on the language of hocolim
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 5, 2015 at 17:35 | comment | added | Ronnie Brown | I get confused by all these ideas since the fundamental groupoid $\pi_1(X)$ of a space $X$ is a strict groupoid, and the version of the Seifert-van Kampen Theorem useful for computation involves $\pi_1(X,C)$, the fundamental groupoid on a set of $C$ of base points. The so called "higher groupoids" of various writers are not groupoids at all, but lax versions, so the analogy is not clear. Another source of confusion is that homotopy groups are defined only for spaces with a given base point. | |
| Jun 24, 2015 at 20:22 | comment | added | Ronnie Brown | Lurie's version is also discussed at mathoverflow.net/questions/102295/… My answer below explains more on my ideas of such higher Seifert-van Kampen theorems. | |
| Jun 20, 2015 at 10:03 | answer | added | Ronnie Brown | timeline score: 2 | |
| Jun 19, 2015 at 6:21 | vote | accept | Sergei Ivanov | ||
| Jun 18, 2015 at 21:05 | comment | added | David Roberts♦ | Thanks, @Adeel. I was on my phone and in a rush, and didn't do the obvious google search! | |
| Jun 18, 2015 at 19:45 | answer | added | Qiaochu Yuan | timeline score: 10 | |
| Jun 18, 2015 at 13:37 | answer | added | Tim Porter | timeline score: 7 | |
| Jun 18, 2015 at 12:07 | comment | added | AAK | @DavidRoberts, Lurie's version is discussed here: ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem | |
| Jun 18, 2015 at 10:02 | comment | added | Fernando Muro | I totally agree with @EricWofsey Morally, the fundamental infinity groupoid is the space itself, so you find the same thing at both sides of the equation. Brown's version need not be for filtered spaces, I mean, the skeletal filtration is fine and canonical. I'd say that such a result really represents a simplification when you replace $\Pi_1$ with something which is easy enough, such that the fundamental crossed module, categorical group, etc. | |
| Jun 18, 2015 at 9:50 | comment | added | David Roberts♦ | I thought Lurie had a version of this, but I recall offhand where. | |
| Jun 18, 2015 at 9:26 | history | edited | Eric Wofsey |
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| Jun 18, 2015 at 9:25 | comment | added | Eric Wofsey | This is closely related to the homotopy hypothesis, and some definitions of higher groupoids make it essentially a tautology. Of course, this is not much use for computations unless you have a definition of higher groupoid from which you can actually compute the homotopy groups of a colimit. | |
| Jun 18, 2015 at 8:45 | history | asked | Sergei Ivanov | CC BY-SA 3.0 |