Matrix invariants of Spectral categories

Abstract

In this paper we pursue the study of spectral categories initiated in [26]. More precisely, we construct the Universal matrix invariant of spectral categories, i.e. a functor U with values in an additive category Add, which inverts the Morita equivalences, satisfies matrix invariance, and is universal with respect to these two properties. For example, the algebraic K-theory and the topological Hochschild and cyclic homologies are matrix invariants, and so they factor uniquely throw U. As an application, we obtain for free non-trivial trace maps from the Grothendieck group to the topological Hochschild homology ones.