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

TL;DR

This paper constructs a new stable $mbda$-category framework for equivariant $KK$-theory, enabling advanced analysis of $C^*$-algebras with group actions and establishing universal properties and functorial relations.

Contribution

It introduces a novel stable $mbda$-category for equivariant $KK$-theory, with universal properties, change-of-group functors, and extensions to $C^*$-categories, advancing the theoretical foundation.

Findings

Constructed a small, idempotent complete, symmetric monoidal, stable $mbda$-category $ ext{KK}^{G}_{ ext{sep}}$.

Established universal properties and functorial relations for the new categories.

Defined a spectrum-valued equivariant $K$-homology theory for proper $G$-spaces.

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.