$\require{AMScd}$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.

  • $\begingroup$ 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

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.


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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