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

We define a class of motivic equivalences of small stable $\mathrm{\infty }$-categories $W_{\mathrm{m}\mathrm{o}\mathrm{t}}$ and show that the Dwyer–Kan localization functor ${\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }}^{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}\to {\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }}^{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}[W_{\mathrm{m}\mathrm{o}\mathrm{t}}^{-1}]$ is the universal localizing invariant in the sense of Blumberg–Gepner–Tabuada. In particular, we show that every object in its target ${\mathcal{M}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ can be represented as ${\mathcal{U}}_{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathcal{C})$ for some small stable $\mathrm{\infty }$-category $\mathcal{C}$. As another consequence, and using work of Efimov, we improve the universal property of ${\mathcal{M}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ and show that any ${\mathrm{\aleph }}_1$-finitary localizing invariant factors uniquely through it.