Abstract

We provide a universal characterization of the construction taking a scheme $X$ to its stable $\infty$-category $Mot(X)$ of noncommutative motives, patterned after the universal characterization of algebraic K-theory due to Blumberg, Gepner and 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 independent interest.

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