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Let $\mathcal{D}: C\to Top$ be a diagram of spaces (spaces are "nice", and $C$ is small). Let $X$ denote the homotopy colimit of $\mathcal{D}$ (which is connected) and $\pi(C)$ be the free groupoid on $C$ (i.e., fundamental groupoid of the geometric realization).

Is it possible to characterize (or express) connected covers of $X$ in terms of (connected) groupoid covers of $\pi(C)$?

For example, if $C$ is acyclic and $\mathcal{D}$ assigns points then it can be done using the Groethendieck construction.

Is there a general recipe that explains such a construction? Can it be done for diagrams of categories ?

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I have not quite thought this through but there are some facts which seem relevant:

  1. For suitably nice $X$, the fundamental groupoid functor gives an equivalence of categories from covering maps over $X$, to covering groupoids over $\pi_1X$; this is proved in my book "Topology and groupoids" (2006) and was in the (differently titled) 1988 edition, and in essence way back in the 1968 edition; it gives an essentially base point free approach to covering space theory. One advantage is also that a map of spaces is modelled by a morphism of groupoids, instead of an action. Lifting morphisms is easier to follow than lifting actions, at least for me.

  2. If $p: Y \to X$ is a map in the category of compactly generated spaces, then the pullback functor $p^*: Top/X \to Top/Y$ has a right adjoint and so preserves colimits. (This is one aspect of the exponential law in spaces over $X$, largely due to Peter Booth, see his entry on MathSciNet.)

  3. If $q: H \to G$ is a fibration of groupoids, then the pullback functor $p^*: Gpd/G \to Gpd/H$ has a right adjoint and so preserves colimits. (This is part of the restricted locally closed aspect of the category of groupoids, which is due to F. Conduch\'e, and even earlier to J. Giraud: in short, the middle morphism of the supposed exponential law has to be a fibration.)

Of course covering morphisms of groupoids are special cases of fibrations.

I thought homotopy colimits are put together from various kinds of colimits; if so that ought to do the job.

Arguments of this type are used in

R. Brown and R. Street, `Covering morphisms of crossed complexes and of cubical omega-groupoids are closed under tensor product', Cah. Top. G\'eom. Diff. Cat., 52 (2011) 188-208.

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Some indications of how this may be studied are given in Andy Tonks' thesis in 1993 from Bangor. He assumes the spaces are represented by crossed complexes and shows how to calculate the homotopy colimit algebraically. This will not answer your question completely but may help. If the spaces $D(c)$ are Elienberg-Mac Lane spaces then there are interpretations of the idea using `higher generation by subgroups' in the work of Abels and Holz. There are also links with complexes of groups in a similar context.

Added later : the point about crossed complexes is that they ar every near to the chains on the universal covering of a space, but have a bit more homotopy recorded in them. The whole question of the generalisations of the van Kampen theorem come into this. You can find a lot on this in R. Brown, P. J. Higgins and R. Sivera, 2010, Nonabelian Algebraic Topology: Filtered spaces, crossed complexes, cubical homotopy groupoids , volume 15 of EMS Tracts in Mathematics , European Mathematical Society. It includes discussions of Tonks results. For the simple case of a homotopy pushout a paper (R. Brown, E. J. Moore, T. Porter and C. D. Wensley, Crossed complexes, and free crossed resolutions for amalgamated sums and HNN-extensions of groups , Georgian Math. J., 9, (2002), 623 – 644) may be of interest.

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