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

2104.04021Apr 8, 2021Minor revisions and improvements. Accepted for publication in Annals of K-Theory. (Numberings here differ from those in the published version.)

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

Aaron Mazel-Gee, Reuben Stern

K-Theory and Homology Algebraic Topology Category Theory

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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

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\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.

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