$\require{AMScd}$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.

noncommutative motives. (@EldenElmanto the stabilization is always linear functors from $S^{fin}_*$, no presentability assumptions needed) $\endgroup$ – Denis Nardin Aug 25 '16 at 15:28