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.

Theorems & Definitions (136)

• Definition 1.1
• Definition 1.2
• Example 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 1.7
• proof
• Theorem 1.8: Theorem \ref{['thm:universal_omega_1_finitary']}
• Theorem 1.9: Theorem \ref{['thm:countable_products']}