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edit approved Feb 13, 2015 at 5:43

Equivariant model structure on $G$$G-$Gpd$\mathrm{Gpd}$

Let's denote $G$-$Gpd$$G\text{-}\mathrm{Gpd}$ the presheaf category $[BG, Gpd]$$[\mathbf{B}G, \mathrm{Gpd}]$. Now assume that $Gpd$$\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 $G$-$Gpd$$G\text{-}\mathrm{Gpd}$ where a map in $G$-$Gpd$$G\text{-}\mathrm{Gpd}$ is a weak equivalence (resp. a fibration) iffif 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 here Defin Definition 3.7 http://www.math.uiuc.edu/~bertg/EquivModels.pdfhere .

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 $Cat$$\mathrm{Cat}$.

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

Thanks

Equivariant model structure on $G$-$Gpd$

Let's denote $G$-$Gpd$ the presheaf category $[BG, Gpd]$. Now assume that $Gpd$ is endowed with its natural model structure where weak equivalences are equivalences of groupoids.

There is a natural candidate for a model structure on $G$-$Gpd$ where a map in $G$-$Gpd$ is a weak equivalence (resp. a fibration) iff 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 here Def 3.7 http://www.math.uiuc.edu/~bertg/EquivModels.pdf .

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 $Cat$.

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

Thanks

Equivariant model structure on $G-\mathrm{Gpd}$

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).

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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asked Feb 12, 2015 at 16:02

user2664

Equivariant model structure on $G$-$Gpd$

Let's denote $G$-$Gpd$ the presheaf category $[BG, Gpd]$. Now assume that $Gpd$ is endowed with its natural model structure where weak equivalences are equivalences 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 here Def 3.7 http://www.math.uiuc.edu/~bertg/EquivModels.pdf .

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 $Cat$.

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

Thanks