A stable $\infty$-category for equivariant $KK$-theory

Abstract

For a countable group $G$ we construct a small, idempotent complete, symmetric monoidal, stable $\infty$-category ${\mathrm{KK}}_{\mathrm{sep}}^G$ whose homotopy category recovers the triangulated equivariant Kasparov category of separable $G$-$C^{\ast }$-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable $\infty$-category ${\mathrm{KK}}^G$ which receives a symmetric monoidal functor ${\mathrm{kk}}^G$ from possibly non-separable $G$-$C^{\ast }$-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying $G$. We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite $K$-homology theory on proper and locally compact $G$-topological spaces, allowing for coefficients in arbitrary $G$-$C^{\ast }$-algebras. Finally, we extend the functor ${\mathrm{kk}}^G$ from $G$-$C^{\ast }$-algebras to $G$-$C^{\ast }$-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps.

108 pages. Minor corrections, References updated