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.
Categories: math.KT math.AT math.CT
