Homotopy theory of G-diagrams and equivariant excision

Abstract

Let $G$ be a finite group acting on a small category $I$. We study functors $X \colon I \to \mathscr{C}$ equipped with families of compatible natural transformations that give a kind of generalized $G$-action on $X$. Such objects are called $G$-diagrams. When $\mathscr{C}$ is a sufficiently nice model category we define a model structure on the category of $G$-diagrams in $\mathscr{C}$. There are natural $G$-actions on Bousfield-Kan style homotopy limits and colimits of $G$-diagrams. We prove that weak equivalences between point-wise (co)fibrant $G$-diagrams induce weak $G$-equivalences on homotopy (co)limits. A case of particular interest is when the indexing category is a cube. We use homotopy limits and colimits over such diagrams to produce loop and suspension spaces with respect to permutation representations of $G$. We go on to develop a theory of enriched equivariant homotopy functors and give an equivariant "linearity" condition in terms of cubical $G$-diagrams. In the case of $G$-topological spaces we prove that this condition is equivalent to Blumberg's notion of $G$-linearity. In particular we show that the Wirthm\"{u}ller isomorphism theorem is a direct consequence of the equivariant linearity of the identity functor on $G$-spectra.