Every motive is the motive of a stable $\infty$-category

Abstract

We define a class of motivic equivalences of small stable $\infty$-categories $W_{\mathrm{mot}}$ and show that the Dwyer--Kan localization functor $\mathrm{Cat}^{\mathrm{perf}}_\infty \to \mathrm{Cat}^{\mathrm{perf}}_\infty[W_{\mathrm{mot}}^{-1}]$ is the universal localizing invariant in the sense of Blumberg--Gepner--Tabuada. In particular, we show that every object in its target $\mathcal{M}_{\mathrm{loc}}$ can be represented as $\mathcal{U}_{\mathrm{loc}}(\mathcal{C})$ for some small stable $\infty$-category $\mathcal{C}$. As another consequence, and using work of Efimov, we improve the universal property of $\mathcal{M}_{\mathrm{loc}}$ and show that any $\aleph_1$-finitary localizing invariant factors uniquely through it.