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$\require{AMScd}$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.

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  • $\begingroup$ Note that in general you don't have a functor $\Sigma^\infty$ from a category to its stabilization, but only a functor $\Omega^\infty$ going in the other direction ($\Sigma^\infty$ exists when the category is presentable by the adjoint functor theorem). Moreover I don't think that the category of spectrally enriched categories is stable. Interesting question though. $\endgroup$ – Denis Nardin Aug 25 '16 at 12:20
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    $\begingroup$ It feels like loops on some pointed category will be the space of automorphisms of the distinguished object, no? So the stabilization should just be spectra back again... To get a more interesting answer one probably needs to at least consider lax pullbacks in your definition of `loops', but I dunno $\endgroup$ – Dylan Wilson Aug 25 '16 at 12:42
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    $\begingroup$ Isn't there some formula for computing stabilizations in terms of presentations? E.g., if C is a localization of Fun(J,Spaces) at some set of maps, then it's stabilization should be a stable localization of Fun(J,Spectra) at a set of maps, no? (I'm asking, I don't really know.) If you compute this on the usual presentation of infty-Cat, you should just get spectra back. $\endgroup$ – Charles Rezk Aug 25 '16 at 13:44
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    $\begingroup$ If you are interested in things that are loosely speaking homology theories on categories, one interesting keyword is 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
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    $\begingroup$ It might be more interesting to think about stabilizations of slices $\infty\mathrm{Cat}_{/C}$. I'd guess that the stabilization of $\infty\mathrm{Cat}_{/\Delta^1}$ is something like spans of spectra. $\endgroup$ – Charles Rezk Aug 25 '16 at 17:39
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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.

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    $\begingroup$ Cool! ................. $\endgroup$ – Charles Rezk Aug 25 '16 at 21:30

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