$KK$-theory spectra for $C^{\ast }$-categories and discrete groupoid $C^{\ast }$-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.