Matrix Invariants of Spectral Categories

In this paper, we further the study of spectral categories initiated in [31]. Our main contribution is the construction of the Universal matrix invariant of spectral categories, that is, a functor $\mathcal U$ with values in an additive category, which inverts the Morita equivalences, satisfies matrix invariance, and is universal with respect to these two properties. For example, algebraic K-theory and topological Hochschild and cyclic homologies are matrix invariants and so they factor (uniquely) through $\mathcal U$. As an application, we obtain a bijective correspondence between the elements of the stable groups of spheres and the trace maps from the Grothendieck group to the topological Hochschild homology groups.