Let M be a monoid and G:Mon→Grp\documentclass[12pt]{minimal}
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\begin{document}$$G:\mathbf {Mon} \rightarrow \mathbf {Grp}$$\end{document} be the group completion functor from monoids to groups. Given a collection X\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {X}$$\end{document} of submonoids of M and for each N∈X\documentclass[12pt]{minimal}
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\begin{document}$$N\in \mathcal {X}$$\end{document} a collection YN\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Y}_N$$\end{document} of subgroups of G(N), we construct a model structure on the category of M-spaces and M-equivariant maps, called the (X,Y)\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {X},\mathcal {Y})$$\end{document}-model structure, in which weak equivalences and fibrations are induced from the standard YN\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Y}_N$$\end{document}-model structures on G(N)-spaces for all N∈X\documentclass[12pt]{minimal}
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\begin{document}$$N\in \mathcal {X}$$\end{document}. We also show that for a pair of collections (X,Y)\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {X},\mathcal {Y})$$\end{document} there is a small category O(X,Y)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbf {O}}}_{(\mathcal {X},\mathcal {Y})}$$\end{document} whose objects are M-spaces M×NG(N)/H\documentclass[12pt]{minimal}
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\begin{document}$$M\times _NG(N)/H$$\end{document} for each N∈X\documentclass[12pt]{minimal}
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\begin{document}$$N\in \mathcal {X}$$\end{document} and H∈YN\documentclass[12pt]{minimal}
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\begin{document}$$H\in \mathcal {Y}_N$$\end{document} and morphisms are M-equivariant maps, such that the (X,Y)\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {X},\mathcal {Y})$$\end{document}-model structure on the category of M-spaces is Quillen equivalent to the projective model structure on the category of contravariant O(X,Y)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbf {O}}}_{(\mathcal {X},\mathcal {Y})}$$\end{document}-diagrams of spaces.
