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Let's denote $G\text{-}\mathrm{Gpd}$ the presheaf category $[\mathbf{B}G, \mathrm{Gpd}]$. Now assume that $\mathrm{Gpd}$ is endowed with its natural model structure where weak equivalences are equivalences of groupoids.

There is a natural candidate for a model structure on $G\text{-}\mathrm{Gpd}$ where a map in $G\text{-}\mathrm{Gpd}$ is a weak equivalence (resp. a fibration) if and only if for every subgroup $H$ of $G$ its restriction to $H$-invariants is a weak equivalence of groupoids (resp. a fibration of groupoids).

A sufficient condition for the existence of this model structure is: the $H$-fixed points functors have to be cellular for every subgroup $H$ of $G$. This condition means that these functors have to preserve some pushouts and some directed colimits. The details on the cellularity condition are available in Definition 3.7 here.

It is well-known that this condtion is satisfied if one starts with the Quillen model structure on simplicial sets. I have also read that it is true for Thomason model structure on $\mathrm{Cat}$.

Is the cellularity condition satisfied if one starts with the natural model structure on $\mathrm{Gpd}$? If yes, where I can find a reference?

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    $\begingroup$ I don't know of a reference but there's a canonical inclusion of $Gpd$ into $sSet$ (by taking the nerve) that preserves weak equivalences and cofibrations and it looks like the colimits you want to study are preserved by this inclusion, so you should be able to backport the result from the Quillen model structure $\endgroup$
    – Denis Nardin
    Commented Feb 12, 2015 at 17:59
  • $\begingroup$ I think you mean that the nerve functor preserves fibrations and trivial fibrations, not "weak equivalences and cofibrations". Right ? Do you have some reference for the preservation of some limits/colimits by the nerve functor ? $\endgroup$
    – user2664
    Commented Feb 12, 2015 at 21:08
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    $\begingroup$ I am sure that the nerve preserves weak equivalences (since it sends them to homotopy equivalences). I checked the "canonical model structure" on groupoids and I discovered that the cofibrations are not what I thought they were (apparently cofibrations are not required to be injections on arrows). I am sorry I do not have references. $\endgroup$
    – Denis Nardin
    Commented Feb 12, 2015 at 23:17
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    $\begingroup$ The model structure on groupoids is transferred from simplicial sets via the nerve functor, which by definition preserves fibrations and weak equivalences. Being a right adjoint it automatically preserves all limits. See Section 2 of Hollander's “A homotopy theory for stacks”, for example. $\endgroup$
    – Dmitri Pavlov
    Commented Feb 13, 2015 at 9:09
  • $\begingroup$ @Dmitri: yes, of course for limits! Thanks. $\endgroup$
    – user2664
    Commented Feb 13, 2015 at 9:37

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