Every motive is the motive of a stable ∞-category

2025/03/14 by Ramzi, Maxime, Sosnilo, Vladimir, Winges, Christoph
#Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2503.11338

Abstract

We define a class of motivic equivalences of small stable ∞-categories W_mot and show that the Dwyer--Kan localization functor Cat^perf_∞ → Cat^perf_∞[W_mot^-1] is the universal localizing invariant in the sense of Blumberg--Gepner--Tabuada. In particular, we show that every object in its target M_loc can be represented as U_loc(C) for some small stable ∞-category C. As another consequence, and using work of Efimov, we improve the universal property of M_loc and show that any ℵ₁-finitary localizing invariant factors uniquely through it.

Related

• Rigidity of the category of localizing motives
• The localisation theorem for the K-theory of stable ∞-categories
• Classification of localizing subcategories along t-structures
• On nilpotent extensions of ∞-categories and the cyclotomic trace
• Key lemma and universal localization