What's the stabilization of the $\infty$-category of $\infty$-categories? $\require{AMScd}$One nice thing about $\infty$-categories is that spaces are themselves $\infty$-categories. What's the analogue for spectra? Presumably this would be the stabilization of the $\infty$-category of $(\infty,1)$-categories, fitting into a cube:
$$
\begin{CD}
ILS_* @>>> \text{Spaces}_* @>>> \text{$(\infty,1)$-Cats}_* @<<< \text{S.M. $(\infty,1)$-Cats}_*\\
@V\simeq VV @V\Sigma^\infty VV @VV\Sigma^\infty V @VV\simeq V\\
\text{ConnSpectra} @>>> \text{Spectra} @>>> ?  @<<< Conn ?
\end{CD}
$$
where the far left/right entries of the top row are respectively symmetric monoidal $(\infty,0/1)$-categories, and the $*$s denote that we have chosen a basepoint. Is this just the $\infty$-category of spectrally enriched categories?
edit if this idea doesn't work out, I'm more interested in "What's the analogue for spectra?" than the second question.
 A: 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.
