Let $G$ be a discrete groupe. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which are equivalences when seen as maps of simplicial sets.

My question is: what are the fibrations of this model category ? (aside from "the maps satisfying the right lifting property...")

This question of what are the fibrations of the injective model structure have already been asked several time on this forum and elsewhere, but it apparently never receive an answer in the more general case. I think this specific case is easier.

Indeed, I've been able to check that the trivial fibrations are exactly the maps $f$ such that for each subgroup $H \subset G$ the map $f$ restricted to a map between $H$-invariants is a trivial kan fibration.

It is hence natural to think that fibrations might be defined by the same property: for each $H$ the restriction of $f$ as a map between the $H$-invariants is a kan fibration.

It is indeed true that fibrations do satisfy this property, but as pointed out by Charles Rezk in the comments, this is not a sufficient condition.

Let $G$ be a discrete group. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which are equivalences when seen as maps of simplicial sets.

My question is: what are the fibrations of this model category ? (aside from "the maps satisfying the right lifting property...")

This question of what are the fibrations of the injective model structure have already been asked several time on this forum and elsewhere, but it apparently never receive an answer in the more general case. I think this specific case is easier.

Indeed, I've been able to check that the trivial fibrations are exactly the maps $f$ such that for each subgroup $H \subset G$ the map $f$ restricted to a map between $H$-invariants is a trivial kan fibration.

It is hence natural to think that fibrations might be defined by the same property: for each $H$ the restriction of $f$ as a map between the $H$-invariants is a kan fibration.

It is indeed true that fibrations do satisfy this property, but as pointed out by Charles Rezk in the comments, this is not a sufficient condition.

Let $G$ be a discrete groupe. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which are equivalences when seen as maps of simplicial sets.

My question is: what are the fibrations of this model category ? (aside from "the maps satisfying the right lifting property...")

This question of what are the fibrations of the injective model structure have already been asked several time on this forum and elsewhere, but it apparently never receive an answer in the more general case. I think this specific case is easier.

Indeed, I've been able to check that the trivial fibrations are exactly the maps $f$ such that for each subgroup $H \subset G$ the map $f$ restricted to a map between $H$-invariants is a trivial kan fibration.

It is hence natural to think that fibrations might be defined by the same property: for each $H$ the restriction of $f$ as a map between the $H$-invariants is a kan fibration.

It is indeed true that fibrations do satisfy this property, but I'm not sure about the converse.

Let $G$ be a discrete groupe. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which are equivalences when seen as maps of simplicial sets.

My question is: what are the fibrations of this model category ? (aside from "the maps satisfying the right lifting property...")

This question of what are the fibrations of the injective model structure have already been asked several time on this forum and elsewhere, but it apparently never receive an answer in the more general case. I think this specific case is easier.

Indeed, I've been able to check that the trivial fibrations are exactly the maps $f$ such that for each subgroup $H \subset G$ the map $f$ restricted to a map between $H$-invariants is a trivial kan fibration.

It is hence natural to think that fibrations might be defined by the same property: for each $H$ the restriction of $f$ as a map between the $H$-invariants is a kan fibration.

It is indeed true that fibrations do satisfy this property, but as pointed out by Charles Rezk in the comments, this is not a sufficient condition.