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 intependent interest.

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