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**On equivariant homotopy theory for model categories.**
*(English)*
Zbl 1357.18005

This article is a contribution to existence and equivalence of model structures in equivariant homotopy theory.

Let us remind a few definitions. If \(\mathcal{C}\) is a model category and \(\mathcal{A}\) a small category, we say that the functor category \(\mathcal{C}^\mathcal{A}\) admits the projective model structure if it has a (necessarily unique) model structure in which fibrations and weak equivalences are defined pointwise from the model structure of \(\mathcal{C}\). If \(G\) is a group (that we can see as a category with one object) and \(\mathcal{F}\) a family of subgroups of \(G\), we say that the category \(\mathcal{C}^G\) of \(G\)-objects in \(\mathcal{C}\) admits the \(\mathcal{F}\)-model structure if it has a (necessarily unique) model structure in which fibrations and weak equivalences are maps which have this property in \(\mathcal{C}\) when applying the functor of \(H\)-invariants for each \(H\in\mathcal{F}\). For that matter we denote by \(\mathcal{O}_\mathcal{F}\) (the orbit category associated to \(\mathcal{F}\)) the full subcategory of \(G\)-sets with objects \(G/H\) for \(H\in\mathcal{F}\).

In its {Theorem 2.10}, the article proves that, if \(\mathcal{C}\) is a cofibrantly generated model category and \(\mathcal{F}\) a family of subgroups of \(G\) containing the trivial one and if moreover, for each \(H\in\mathcal{F}\), the functor of \(H\)-invariants satisfies some cellularity conditions which are stated in Proposition 2.6 (the first of these conditions, for example, is that it preserves filtered colimits of cofibrations), then:

(i) \(\mathcal{C}^{\mathcal{O}_\mathcal{F}^{op}}\) admits the projective model structure;

(ii) \(\mathcal{C}^G\) admits the \(\mathcal{F}\)-model structure;

(iii) these model categories are Quillen equivalent.

As the article explains, it allows to recover several known results in particular cases. It allows also to get a conceptual generalization of the main result of [P.H. Kropholler and C.T.C. Wall, Publ. Mat., Barc. 55, No. 1, 3–18 (2011; Zbl 1216.55007)].

In its third part, the article gives a result (Theorem 3.17) which is similar to Theorem 2.10 in a topological setting: \(G\) is a compact Lie group, \(\mathcal{F}\) a family of closed subgroups and \(\mathcal{C}\) a topological model category.

Let us remind a few definitions. If \(\mathcal{C}\) is a model category and \(\mathcal{A}\) a small category, we say that the functor category \(\mathcal{C}^\mathcal{A}\) admits the projective model structure if it has a (necessarily unique) model structure in which fibrations and weak equivalences are defined pointwise from the model structure of \(\mathcal{C}\). If \(G\) is a group (that we can see as a category with one object) and \(\mathcal{F}\) a family of subgroups of \(G\), we say that the category \(\mathcal{C}^G\) of \(G\)-objects in \(\mathcal{C}\) admits the \(\mathcal{F}\)-model structure if it has a (necessarily unique) model structure in which fibrations and weak equivalences are maps which have this property in \(\mathcal{C}\) when applying the functor of \(H\)-invariants for each \(H\in\mathcal{F}\). For that matter we denote by \(\mathcal{O}_\mathcal{F}\) (the orbit category associated to \(\mathcal{F}\)) the full subcategory of \(G\)-sets with objects \(G/H\) for \(H\in\mathcal{F}\).

In its {Theorem 2.10}, the article proves that, if \(\mathcal{C}\) is a cofibrantly generated model category and \(\mathcal{F}\) a family of subgroups of \(G\) containing the trivial one and if moreover, for each \(H\in\mathcal{F}\), the functor of \(H\)-invariants satisfies some cellularity conditions which are stated in Proposition 2.6 (the first of these conditions, for example, is that it preserves filtered colimits of cofibrations), then:

(i) \(\mathcal{C}^{\mathcal{O}_\mathcal{F}^{op}}\) admits the projective model structure;

(ii) \(\mathcal{C}^G\) admits the \(\mathcal{F}\)-model structure;

(iii) these model categories are Quillen equivalent.

As the article explains, it allows to recover several known results in particular cases. It allows also to get a conceptual generalization of the main result of [P.H. Kropholler and C.T.C. Wall, Publ. Mat., Barc. 55, No. 1, 3–18 (2011; Zbl 1216.55007)].

In its third part, the article gives a result (Theorem 3.17) which is similar to Theorem 2.10 in a topological setting: \(G\) is a compact Lie group, \(\mathcal{F}\) a family of closed subgroups and \(\mathcal{C}\) a topological model category.

Reviewer: Aurelien Djament (Nantes)

### MSC:

18G55 | Nonabelian homotopical algebra (MSC2010) |

55P91 | Equivariant homotopy theory in algebraic topology |

20J99 | Connections of group theory with homological algebra and category theory |