Local coefficients for genuine equivariant cohomology I

Abstract

We develop a theory of equivariant local systems which categorifies genuine equivariant cohomology theories -- such as Atiyah--Segal $K$-theory, Lurie's tempered cohomology, and the equivariant elliptic cohomology of Grojnowski, Greenlees, and Gepner--Meier -- analogously to how the category of ordinary local systems categorifies singular cohomology. More precisely, we introduce, for a global space $X$ and a coefficient system $\mathscr{A} \colon \mathrm{Orb}^{\mathrm{op}} \to \mathrm{Pr}^{\mathrm{L}}$, the categories $\mathrm{LS}^{\mathrm{glo}}(X,\mathscr{A})$ and $\mathrm{LS}^{\mathrm{gen}}(X,\mathscr{A})$ of globally equivariant and genuine local systems, the latter generalizing the genuine stable category $\mathrm{Sp}^G$ of a compact Lie group $G$ to non-constant coefficients. When the coefficients come from an oriented abelian group stack $A$ over a locally complex periodic base, we also construct the category $\mathrm{LS}^\mathrm{temp}(X,A)$ of tempered local systems, extending Lurie's theory beyond finite groups; the defining condition is justified by a tempered form of the Atiyah--Segal completion theorem. We show that global sections induce an equivalence $\mathrm{LS}^{\mathrm{temp}}(BU(1),A) \simeq \mathrm{QCoh}(A)$ and we prove that $\mathrm{LS}^\mathrm{temp}(X,A)$ is a smashing localization of $\mathrm{LS}^\mathrm{gen}(X,A)$ if $X$ is an orbispace.