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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4$\begingroup$ Consider for simplicity a spectrum $X$ with $\pi_k(X)=0$ for $k\leq 0$. Then the KanThurston 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 KanThurston to prove your conjecture, but that it would be difficult to deduce it from KanThurston. $\endgroup$ – Neil Strickland Dec 29 '16 at 17:50

1$\begingroup$ 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 '16 at 20:05