An ElmendorfPiacenza type Theorem for Actions of Monoids
Abstract
Let $M$ be a monoid and $G:\mathbf{Mon} \to \mathbf{Grp}$ be the group completion functor from monoids to groups. Given a collection $\mathcal{Z}$ of submonoids of $M$ and for each $N\in \mathcal{Z}$ 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, in which weak equivalences and fibrations are determined by the standard $\mathcal{Y}_N$model structures on $G(N)$spaces for all $N\in \mathcal{Z}$. We also show that there is a small category $\mathsf{O}_(\mathcal{Z},\mathcal{Y})$ such that, under mild conditions on $\mathcal{Z}$ and $\mathcal{Y}_N$'s, the projective model structure on the category of contravariant $\mathsf{O}_(\mathcal{Z},\mathcal{Y})$diagrams of spaces is Quillen equivalent to our model structure. In particular, we prove a theorem generalizing ElmendorfPiacenza Theorem to actions of monoids.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.06785
 Bibcode:
 2016arXiv160906785A
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Algebraic Topology;
 Mathematics  Dynamical Systems;
 55U35;
 55U40
 EPrint:
 22 pages, some notations and title changed, new lemmas/propositions added and some old results removed