A stable $\infty$-category for equivariant $KK$-theory
Ulrich Bunke, Alexander Engel, Markus Land
TL;DR
The paper constructs and analyzes stable ∞-categorical frameworks for equivariant KK-theory, introducing the small and presentable KK categories ${ m KK}^{G}_{ m sep}$ and ${ m KK}^{G}$ with universal properties. It develops an equivariant, spectrum-valued analytic K-homology $K^{G, m an}$ and proves key exactness, homotopy-invariance, and locality properties, enabling coefficients in arbitrary $G$-C*-algebras and extension to $G$-C*-categories. Change-of-groups functors, Green–Julg-type dualities, and descent results are formulated at the ∞-categorical level, alongside a comprehensive treatment of tensor products for $C^{*}$-categories. The framework provides a robust foundation for applications to Paschke duality and assembly maps, and yields a spectrum-valued refinement of equivariant K-homology compatible with various coefficient and categorical enhancements.
Abstract
For a countable group $G$ we construct a small, idempotent complete, symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}_{\mathrm{sep}}$ whose homotopy category recovers the triangulated equivariant Kasparov category of separable $G$-$C^*$-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 $G$-$C^*$-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 $G$-$C^*$-algebras. Finally, we extend the functor $\mathrm{kk}^{G}$ from $G$-$C^*$-algebras to $G$-$C^*$-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps.