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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