Matrix invariants of Spectral categories

TL;DR

This paper develops a universal matrix invariant for spectral categories that captures key invariants like algebraic K-theory and Hochschild homology, enabling new trace maps.

Contribution

It constructs the universal matrix invariant functor for spectral categories, unifying various invariants and enabling new applications.

Findings

Universal matrix invariant functor U constructed

U inverts Morita equivalences and satisfies matrix invariance

Facilitates new trace maps from Grothendieck group to Hochschild homology

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.