\(KK\)-theory of \(C^*\)-categories and the analytic assembly map (Q1866110)

There are a number of different maps referred to as assembly maps, and in some cases several apparently different definitions exist for a given assembly map. It is known (Weiss, Williams, Davis, Luck) that a given assembly map which can be expressed as a map of spectra rather than simply as a group homomorphism is characterised by the fact that it maps something computable to the object of interest in a natural way. The main purpose of the paper is to define \(KK\)-theory spectra associated to \(C^*\)-categories and look at certain instances of the Kasparov product at this level. This new machinery is used to give a description of the analytic assembly map as a natural map of spectra.