Let M be a monoid and \(G:\mathbf {Mon} \rightarrow \mathbf {Grp}\) be the group completion functor from monoids to groups. Given a collection \(\mathcal {X}\) of submonoids of M and for each \(N\in \mathcal {X}\) a collection \(\mathcal {Y}_N\) of subgroups of G(N), we construct a model structure on the category of M-spaces and M-equivariant maps, called the \((\mathcal {X},\mathcal {Y})\)-model structure, in which weak equivalences and fibrations are induced from the standard \(\mathcal {Y}_N\)-model structures on G(N)-spaces for all \(N\in \mathcal {X}\). We also show that for a pair of collections \((\mathcal {X},\mathcal {Y})\) there is a small category \({{\mathbf {O}}}_{(\mathcal {X},\mathcal {Y})}\) whose objects are M-spaces \(M\times _NG(N)/H\) for each \(N\in \mathcal {X}\) and \(H\in \mathcal {Y}_N\) and morphisms are M-equivariant maps, such that the \((\mathcal {X},\mathcal {Y})\)-model structure on the category of M-spaces is Quillen equivalent to the projective model structure on the category of contravariant \({{\mathbf {O}}}_{(\mathcal {X},\mathcal {Y})}\)-diagrams of spaces.

设M是一个幺半群，\(G:\mathbf {Mon} \rightarrow \mathbf {Grp}\)是从幺半群到群的群完成函子。给定M的子类群的集合\(\mathcal {X}\)和对于每个\(N\in \mathcal {X}\) G ( N )的子群的集合\(\mathcal {Y}_N\ ) ，我们在M -空间和M -等变映射的类别上构建模型结构，称为\((\mathcal {X},\mathcal {Y})\) -模型结构，其中诱导弱等价和纤维化来自标准\(\mathcal {Y}_N\) -模型结构所有\(N\in \mathcal {X}\)的G ( N )-空间。我们还表明，对于一对集合\((\mathcal {X},\mathcal {Y})\)有一个小类别\({{\mathbf {O}}}_{(\mathcal {X} ,\mathcal {Y})}\ )对于每个\( N \in \ mathcal {X}\)和\(H\in \ mathcal {Y}_N\)和态射是M -等变映射，使得 M -空间范畴上的\ ((\mathcal {X},\mathcal {Y})\) -模型结构是 Quillen 等价于逆变范畴的投影模型结构\({{\mathbf {O}}}_{(\mathcal {X},\mathcal {Y})}\) -空间图。