In a project in progress with Matan Prasma and Joost Nuiten concerning the abstract cotangent complex formalism we compute the stabilization of the $\infty$-category $\infty\mathrm{Cat}_{/C}$ of $\infty$-categories over a fixed $\infty$-category $C$, and show that it is equivalent to the $\infty$-category of functors $\mathrm{Tw}(C) \to \mathrm{Spectra}$ from the twisted arrow category of $C$ to spectra. In particular, the stabilization of $\infty\mathrm{Cat}$ is equivalent to the $\infty$-category of spectra and the stabilization of $\infty\mathrm{Cat}_{/\Delta^1}$ is equivalent to the $\infty$-category of spans of spectra, as suggested by Charles Rezk in the remarks above.

Another way of phrasing it is to say that the data of a spectrum object in $\infty\mathrm{Cat}_{/C}$ is equivalent to the data of decorating, for each $x,y \in C$, the mapping space $Map_C(x,y)$ with a parametrized spectrum over it, in a way that is suitably functorial in $(x,y) \in C^{op} \times C$. In this language one can phrase and prove the result in the more general setting of enriched categories. We expect to upload this preprint to the arXiv very soon.

In a project in progress with Matan Prasma and Joost Nuiten concerning the abstract cotangent complex formalism we compute the stabilization of the $\infty$-category $\infty\mathrm{Cat}_{/C}$ of $\infty$-categories over a fixed $\infty$-category $C$, and show that it is equivalent to the $\infty$-category of functors $\mathrm{Tw}(C) \to \mathrm{Spectra}$ from the twisted arrow category of $C$ to spectra. In particular, the stabilization of $\infty\mathrm{Cat}$ is equivalent to the $\infty$-category of spectra and the stabilization of $\infty\mathrm{Cat}_{/\Delta^1}$ is equivalent to the $\infty$-category of (co)spans of spectra, as suggested by Charles Rezk in the remarks above.

Another way of phrasing it is to say that the data of a spectrum object in $\infty\mathrm{Cat}_{/C}$ is equivalent to the data of decorating, for each $x,y \in C$, the mapping space $Map_C(x,y)$ with a parametrized spectrum over it, in a way that is suitably functorial in $(x,y) \in C^{op} \times C$. In this language one can phrase and prove the result in the more general setting of enriched categories. We expect to upload this preprint to the arXiv very soon.

Edit:

The paper is now on the arXiv and can be found here.