On the homotopy fixed points of Maurer–Cartan spaces with finite group actions

Abstract

We develop the basic theory of Maurer–Cartan simplicial sets associated to (shifted complete) $L_{\mathrm{\infty }}$-algebras equipped with the action of a finite group. Our main result asserts that the inclusion of the fixed points of this equivariant simplicial set into the homotopy fixed points is a homotopy equivalence of Kan complexes, provided the $L_{\mathrm{\infty }}$-algebra is concentrated in nonnegative degrees. As an application, and under certain connectivity assumptions, we provide rational algebraic models of the fixed and homotopy fixed points of mapping spaces equipped with the action of a finite group.