Timeline for Grothendieck fibrations and classifying spaces
Current License: CC BY-SA 3.0
11 events
when toggle format | what | by | license | comment | |
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May 16, 2013 at 21:03 | comment | added | Ronnie Brown | @David: The facts on crossed complexes are in the EMS Tract Vol 15 on "Nonabelian algebraic topology" (2011) advertised on my web site (with pdf) and the EMS web site. | |
May 16, 2013 at 20:59 | comment | added | Ronnie Brown | @David: R.Brown, "Fibrations of groupoids", J. Algebra, 15 (1970) 103-132, gave the first definition, and a paper by Anderson, Bull AMS, 1978 contains the facts on geometric realisations you might need. I use the exact sequence of a fibration of groupoids in my book "Topology and groupoids", which also has a Mayer-Vietoris type sequence in the chapter on covering spaces. Maybe also P.R. Heath, groupoid operations and fibre homotopy equivalences, Math Z. 130 (1973) 207-233, is relevant to your interests. | |
May 16, 2013 at 17:39 | comment | added | David Carchedi | @Ronnie: You can still tell me the reference though, for "general knowledge". Thanks! | |
May 16, 2013 at 17:26 | comment | added | David Carchedi | @Ronnie: Thanks. Unfortunately, I have a specific goal in mind, and in the case I care about, $\mathcal{C}$ is not a groupoid and has a very interesting homotopy type. | |
May 16, 2013 at 17:19 | comment | added | Ronnie Brown | I mention that that the case $D,C$ are groupoids is well studied, as special cases of fibrations of crossed complexes, although the groupoid case is simpler. Does that help? References given if needed. | |
May 16, 2013 at 16:21 | answer | added | David Carchedi | timeline score: 2 | |
May 15, 2013 at 20:53 | comment | added | David Carchedi | (However, I'm not sure how computationally tractable this is, e.g. for computing homotopy groups, cohomology groups, etc.) | |
May 15, 2013 at 20:50 | comment | added | David Carchedi | Yes, Benjamin, thanks. That is right. That is certainly one result in this direction. This also implies that $\BC$ is the homotopy colimit of constant functor from $\mathcal{C}^{op}$ to spaces, with value the terminal object. | |
May 15, 2013 at 19:03 | comment | added | Benjamin Steinberg | I'm totally not an expert on this, so I may be saying nonsense but doesn't one have by a result of Thomason that BD is homotopy equivalent to a homotopy colinit of the classify spaces of these groupoids induced by the action of C, or something like that? | |
May 15, 2013 at 17:51 | comment | added | Dylan Wilson | Related/generalization: What can be said about the geometric realization of a left fibration of simplicial sets? | |
May 15, 2013 at 17:34 | history | asked | David Carchedi | CC BY-SA 3.0 |