$KK$-theory spectra for $C^\ast$-categories and discrete groupoid $C^\ast$-algebras

TL;DR

This paper develops a refined bivariant K-theory framework using symmetric spectra to represent KK-theory for C*-categories and discrete groupoid C*-algebras, enabling a smash product formulation of the Kasparov product.

Contribution

It introduces a new spectral model for KK-theory in the context of C*-categories and groupoid C*-algebras, extending Cuntz's approach.

Findings

Spectral models for KK-theory are constructed.

Kasparov product is expressed as a smash product of spectra.

Framework applies to both C*-categories and discrete groupoid C*-algebras.

Abstract

In this paper we refine a version of bivariant $K$-theory developed by Cuntz to define symmetric spectra representing the $KK$-theory of $C^\ast$-categories and discrete groupoid $C^\ast$-algebras. In both cases, the Kasparov product can be expressed as a smash product of spectra.