A universal characterization of noncommutative motives and secondary algebraic K-theory

We provide a universal characterization of the construction taking a scheme X to its stable ∞-category Mot(X) of noncommutative motives, patterned after the universal characterization of algebraic K-theory due to Blumberg-Gepner-Tabuada.As a consequence, we obtain a corepresentability theorem for secondary K-theory.We envision this as a fundamental tool for the construction of trace maps from secondary K-theory.Towards these main goals, we introduce a preliminary formalism of "stable (∞, 2)-categories"; notable examples of these include (quasicoherent or constructible) sheaves of stable ∞-categories.We also develop the rudiments of a theory of presentable enriched ∞-categories -and in particular, a theory of presentable (∞, n)-categories -which may be of independent interest.2.2 Additive invariants . . . . . . . .