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For posterity I'll remark that this seems to be discussing the wrong notion of hypergroup. There are two notions: one involving statistics and probability, and the other being an entirely discrete and algebraic idea involving an operation on a set that takes pairs of elements to subsets of that set.
But I think there's another wrinkle here. The delooping of, say, a hypergroup, which is a kind of 𝔽₁-module, isn't going to be a category at all. So you actually need some kind of generalization of (∞,n)-category, I think. But this is all just wild speculation.
@Z.M well, I'm not intentionally trying to say anything about symmetric monoidal categories, that I'm aware of. For instance, the delooping of a commutative monoid is just an ordinary category.
Additionally, Connes and Consani don't talk about functors from $\Gamma^{op}$ into $Top$ or $sSet$ at all, so that's another reason to avoid calling them $E_\infty$-monoids. Commutative monoids are a special class of $\mathbb{F}_1$-modules. And if you want to talk about, say, $\mathbb{F}_1$-spectra, you're going to need higher categories.
Note that Connes and Consani don't put any conditions on the functors out of $\Gamma^{op}$. If they asked for them to be what Segal called "special" then they'd get commutative monoids in $Set$. But the don't.
Consider that the delooping of a monoid should not be a space, but a category. And then the delooping of that should not be a category but a 2-category. So in general if you want the Eilenberg-MacLane spectrum of a monoid, or some other $\mathbb{F}_1$-module, you'll need higher categories.
@Z.M Two comments: First, there aren't any $E_\infty$-monoids here, just $\Gamma$-sets. Second, the $(\infty,n)$-categories are only needed if one wants to start delooping $\mathbb{F}_1$-modules.
