Matrix invariants of spectral categories

In this paper, we further the study of spectral categories initiated in [31]. Our main contributionis the construction of the universal matrix invariant of spectral categories,that is, a functor 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 arematrix invariants and so they factor (uniquely) through U. As an application, we obtaina bijective correspondence between the elements of the stable groups of spheres andthe trace maps from the Grothendieck group to the topological Hochschild homology groups.