Dg categories of cubic fourfolds

We prove a reconstruction theorem \`a la Calabrese-Groechenig for the moduli space parametrizing skyscraper sheaves on a smooth projective variety when these are considered as a system of points in the dg category of perfect complexes on the variety, as axiomatized by To\"en and Vaqui\'e. This result is then used to show that, for a cubic fourfold $Y\subset \mathbb{P}^5_\mathbb{C}$, the Kuznetsov category $\mathcal{A}_Y$ is geometric (possibly twisted) if and only if a dg enhancement $T_Y$ of $\mathcal{A}_Y$ admits a system of points whose associated moduli space is a (possibly twisted) K3 surface.