The authors provide a universal characterization of the construction that maps a scheme \(X\) to its stable \(\infty\)-category \(\text{Mot}(X)\) of noncommutative motives. This construction is inspired by the universal characterization of algebraic K-theory developed by Blumberg, Gepner, and Tabuada. The paper’s primary contribution is a corepresentability theorem for secondary K-theory, which the authors envision as a fundamental tool for constructing trace maps from secondary K-theory. To achieve this, they introduce a preliminary formalism of “stable \((\infty, 2)\)-categories”, with notable examples including quasicoherent or constructible sheaves of stable \(\infty\)-categories. Additionally, Mazel-Gee and Stern develop the rudiments of a theory of presentable enriched \(\infty\)-categories, and more specifically, a theory of presentable \((\infty, n)\)-categories which may be of independent interest. The paper is structured as follows:

1.

Introduction: Overview, background, motivation, main results, questions, and further directions.

2.

Stable \(2\)-categories and localization sequences: Definitions and properties of stable \(2\)-categories, 1-localization sequences, and \(2\)-localization sequences.

3.

From motives to K-theory: Discussion on motives, additive invariants, and the transition from motives to K-theory.

4.

From \(2\)-motives to secondary K-theory: Exploration of \((2,1)\)-motives, \((2,1)\)-additive invariants, and the transition from \((2,1)\)-motives to \((2,1)\)-ary K-theory and secondary K-theory.

5.

Two appendices about enriched category theory and higher category theory.