$ \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Vscr{\mathcal{V}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\we{\simeq} \newcommand\qc{\mathrm{qc}} \newcommand\id{\mathrm{id}} \newcommand\Sp{\mathrm{Sp}} \newcommand\Cat{\mathrm{Cat}} \newcommand\perf{\mathrm{perf}} \newcommand\Mot{\mathrm{Mot}} \newcommand\loc{\mathrm{loc}} \newcommand\unit{\mathbf{1}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Fun{\mathrm{Fun}} \newcommand\fin{\mathrm{fin}} \newcommand\Ac{\mathrm{Ac}} $

Motiquity means the ubuquity of motives.

In their paper [2], which I wrote about here, Ramzi–Sosnilo–Winges proved that every spectrum is the $\K$-theory spectrum of a stable $\infty$-category. Ramzi, Sosnilo, and Winges have posted [3], which proves a fantastic result about the $\infty$-category $\Mot$ of noncommutative motives in the sense of [1] and in particular gives a new proof of the theorem from the previous paper.

Theorem. If $\M\colon\Cat_\infty^\perf\rightarrow\Mot$ denotes the universal finitary localizing invariant, then $\M$ is essentially surjective. More precisely, $\Mot$ is a Dwyer–Kan localization of $\Cat_\infty^\perf$ at the class of morphisms $W$ consisting of those $f$ such that $E(f)$ is an equivalence for every finitary localizing invariant.

Moreover, using work of Efimov, $\Mot$ agrees with the Dwyer–Kan localization at the class of morphisms $f$ such that $\K(\Cscr\otimes(-))$ is an equivalence.

Methods. The basic idea is to use a construction of Grayson to obtain suspensions for stability and to use. Another is to prove that $\Cat_\infty^\perf$ admits the structure of a cofibration category in Cisinski’s sense.

Applications. One notable application is that $\M$ preserves countable products. Another is that $\M$ admits another universal property as the universal $\aleph_1$-finitary localizing invariant, which means in particular that $\TC$ induces a functor $\Mot\rightarrow\Sp$.

Coming to you live from the Abel Symposium meeting, Ramzi explained another application based on forthcoming joint work with Kaif Hilman. Specifically, if $G$ is a finite group and $\underline{\Mot}$ denotes the functor $\underline{\Mot}(G/H)=\Mot((\Cat_\infty^\perf)^{\B H})$, then $\underline{\Mot}$ admits the structure of a $G$-symmetric monoidal $G$-category.

Bonus material. Details are given on Grayson’s construction as well as the variant where one works with motives relative to some other symmetric monoidal $\infty$-category $\Vscr$.

References

[1] Blumberg, Gepner, Tabuada, A universal characterization of higher algebraic $\K$-theory, Geometry & Topology 17 (2013), 733–838.

[2] Ramzi, Sosnilo, Winges, Every spectrum is the K-theory of a stable $\infty$-category, arXiv:2401.06510.

[3] Ramzi, Sosnilo, Winges, Every motive is the motive of a stable $\infty$-category, arXiv:2503.11338.