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 ${\mathrm{Cat}}_{\infty }^{\mathrm{perf}}\to {\mathrm{Cat}}_{\infty }^{\mathrm{perf}}[W_{\mathrm{mot}}^{-1}]$ 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.

31 pages, comments welcome!