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

Abstract

We provide a universal characterization of the construction taking a scheme $X$ to its stable $\infty$-category $\text{Mot}(X)$ 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 $(\infty, 2)$-categories"; notable examples of these include (quasicoherent or constructible) sheaves of stable $\infty$-categories. We also develop the rudiments of a theory of presentable enriched $\infty$-categories -- and in particular, a theory of presentable $(\infty, n)$-categories -- which may be of intependent interest.