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 $\mathcal{A}:{\mathrm{Orb}}^{\mathrm{op}}\to {\mathrm{Pr}}^{\mathrm{L}}$, the categories ${\mathrm{LS}}^{\mathrm{glo}}(X,\mathcal{A})$ and ${\mathrm{LS}}^{\mathrm{gen}}(X,\mathcal{A})$ 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.

129 pages. Comments very welcome!