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Thomason showed that symmetric monoidal categories model all connective spectra. But can it be done with groupoids? In order for this to be the case, the group completion prices process must be very drastic. But after all, the sphere spectrum is the group completion of the symmetric monoidal groupoid of finite sets, so maybe anything is possible.

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    $\begingroup$ Consider for simplicity a spectrum $X$ with $\pi_k(X)=0$ for $k\leq 0$. Then the Kan-Thurston Theorem says that there is a discrete group $G$ such that applying Quillen's plus construction to $BG$ gives $\Omega^\infty X$. The plus construction is at least related to group completion. My guess is that you could adapt the ideas behind Kan-Thurston to prove your conjecture, but that it would be difficult to deduce it from Kan-Thurston. $\endgroup$ Dec 29, 2016 at 17:50
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    $\begingroup$ Thanks! The citation trail leads to this paper of Pirashvili which seems to (almost?) answer my question in the affirmative, but I'm a little confused because what he really shows is that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-group -- so it seems that he's one delooping away from talking about the same realization functor I have in mind... $\endgroup$
    – Tim Campion
    Dec 29, 2016 at 20:05

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Theorem 5.3 of

Daniel Fuentes-Keuthan, Modelling Connective Spectra via Multicategories, arXiv:1909.11148.

answers this question positively!

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    $\begingroup$ Hooray! That's a fantastic result. $\endgroup$
    – Tim Campion
    May 2, 2021 at 19:32

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