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Suppose that $$F:\mathcal{D} \to \mathcal{C}$$ is a Grothendieck fibration of small categories, whose fibers are groupoids. Is there anything sensible which can be said about the induced morphism $$BF:B \mathcal{D} \to B\mathcal{C}$$ between classifying spaces? It seems that Quillen's theorems do not buy us much when the fibers are groupoids, since fibers are homotopy equivalent if and only if they are equivalent as groupoids. I also care about the special case of discrete fibrations (i.e. the Grothendieck construction of a presheaf of sets). Is there some way of expressing the homotopy type of $B \mathcal{D}$ in terms of the homotopy type of $B\mathcal{C}$ together with the data of the psuedofunctor $$\varphi:\mathcal{C}^{op} \to \mathcal{Gpd}$$ classified by the fibration $F,$ which is computationally tractable, at least for "nice enough" $\varphi$? I am aware of a spectral sequence involving the homology of a $\mathcal{C}$-module derived from $\varphi$ converging to the homology of $B\mathcal{D},$ but I would really like something computable from knowing only the homotopy type of $B\mathcal{C},$ not $\mathcal{C}$ itself.

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Related/generalization: What can be said about the geometric realization of a left fibration of simplicial sets? –  Dylan Wilson May 15 '13 at 17:51
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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? –  Benjamin Steinberg May 15 '13 at 19:03
    
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. –  David Carchedi May 15 '13 at 20:50
    
(However, I'm not sure how computationally tractable this is, e.g. for computing homotopy groups, cohomology groups, etc.) –  David Carchedi May 15 '13 at 20:53
    
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. –  Ronnie Brown May 16 '13 at 17:19
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Since others seem to be interested in this question, p. 19 of this preprint http://arxiv.org/abs/1112.3996 appears to have nice spectral sequences for homology (resp.cohomology) of $B\mathcal{D}$ in terms of homology (resp. cohomology) of $B\mathcal{C}$ with coefficients twisted by a local system determined by $\varphi.$

Actually, I am a bit confused. Could anyone tell me why $H^{q}\left(BD_\left(\mspace{3mu} \cdot \mspace{3mu}\right),A\right)$ is a local system? It seems like I would need $\varphi$ to invert all morphisms.

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