Timeline for First mention of the fundamental bigroupoid of a space?
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15 events
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| Mar 24, 2016 at 5:04 | comment | added | David Roberts♦ | @ToddTrimble that's ok: I can just say you are an example of doing it that way. Tom says explicitly in his book 'Higher Operads,...' that is what he is doing, so that's two references which should satisfy a referee. | |
| Mar 24, 2016 at 0:35 | comment | added | Todd Trimble | @DavidRoberts Sorry, I don't really know. | |
| Mar 22, 2016 at 1:27 | comment | added | David Roberts♦ | @ToddTrimble thanks for the tip. As an aside, do you know where weak enrichment over Cat (as a 2-category) was first used to define bicategories? Was it you, even if only in the bigroupoid case? I was pretty sure I had a citation to you in my slow-arriving paper, but it's not there now. I'll put it back in! | |
| Mar 21, 2016 at 12:09 | comment | added | Todd Trimble | What I called the fundamental $n$-groupoid $\Pi_n(X)$ of a space, as part of an inductive notion of $n$-category, was described by me in a talk at Cambridge in 1999 (I first found it in 1996 though); a little later Tom Leinster and Eugenia Cheng both gave written accounts of these definitions. As I'm sure you know there is material on this in the nLab. | |
| Mar 23, 2013 at 1:06 | comment | added | David Roberts♦ | @Ronnie - I'm certainly interested in other variations, just that some are less suited to the constructions I have in mind. For instance, analogous to how the universal covering space can be recovered from $Pi_1$, one can construct from $\Pi_2$ a bundle of topological groupoids which is, suitably interpreted, 2-connected (this was my PhD thesis). My interest in other models is whether such things can be extracted from other higher homotopy groupoids (in particular but not restricted to dimension 2). | |
| Mar 22, 2013 at 12:20 | comment | added | Ronnie Brown | @David: Why not use cubical models? There is a strict double groupoid model, see (R. Brown, H. Kamps and T. Porter), `A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem', TAC, 14 (2005) 200-220.Alternatively, one can take the $n$-truncated cubical singular complex (with connections preferably) and then take homotopy classes rel boundaries at the top level. All the multiple compositions should be quite clear, though I have not written this out. One could also do a Moore version of this. Also one could topologise this as in your paper. | |
| Feb 28, 2013 at 23:16 | answer | added | Ronnie Brown | timeline score: 7 | |
| Feb 28, 2013 at 11:46 | comment | added | David Roberts♦ | Ah, interesting. Given that Grothendieck didn't even stop at 3-groupoids, I should have expected that other people might have considered higher cases. | |
| Feb 28, 2013 at 5:47 | comment | added | Evan Jenkins | math.unice.fr/~cberger/bmonoid.pdf It's one level too high, but this 1997 paper by Berger describes lax 3-groupoids as a model for 3-types; he cites an apparently unpublished 1994 paper of O. Leroy as the original source for this construction. | |
| Feb 27, 2013 at 10:34 | comment | added | David Roberts♦ | @Jonathan: I don't mean general algebraic models of 2-types, but 2-groupoid models. (Note that the full, explicit definition of $\Pi_2$ given by Stevenson takes 5 pages, whereas the crossed module associated to a cellular complex can be defined in a paragraph at most.) | |
| Feb 27, 2013 at 9:59 | comment | added | Jonathan Wise | Whitehead was talking about crossed modules in 1941. | |
| Feb 27, 2013 at 7:27 | comment | added | David Roberts♦ | I was to start with, but now I'm not sure. I'd like any references between 1975/83 (Grothendieck) and 2000 (Stevenson), apart from Baez-Dolan. I was wondering how much it was 'in the air', given the progress on TQFTs and the like in the 1980s-90s. | |
| Feb 27, 2013 at 6:48 | comment | added | Qiaochu Yuan | Are you saying you expect a first mention predating Grothendieck? | |
| Feb 27, 2013 at 6:11 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
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| Feb 27, 2013 at 5:37 | history | asked | David Roberts♦ | CC BY-SA 3.0 |