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

Aaron Mazel-Gee and Reuben Stern

Vol. 9 (2024), No. 2, 369–445
Abstract

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 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 (,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.

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Keywords
K-theory, secondary K-theory, noncommutative motives, universal characterization, stable $(\infty, 2)$-categories
Mathematical Subject Classification
Primary: 18F25, 19D99
Secondary: 18D20, 18N25, 18N65, 55P42
Milestones
Received: 9 January 2024
Revised: 22 January 2024
Accepted: 11 April 2024
Published: 18 August 2024
Authors
Aaron Mazel-Gee
Department of Mathematics
California Institute of Technology
Pasadena, CA
United States
Reuben Stern
Boston, MA
United States