Let G be discrete group and F be a collection of subgroups of G. We show that there exists a left induced model structure on the category of right G-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on H -orbits for all H in F. This gives a model categorical criterion for maps that induce weak equivalences on H -orbits to be weak equivalences in the F-model structure.
Keywords Equivariant homotopy • Orbit space • Model structure
Mathematics Subject Classification 55U40 • 55U35G S of G-spaces, called the F-model structure, in which the weak equivalences and fibrations are maps that induce weak equivalences and fibrations on H -fixed points for Communicated by Emily Riehl.
Abstract. In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gröthendieck group.
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