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I recently noticed the following categorical/universal way to describe the passage from $\mathbb{Z}$ to $\mathbb{Q}$:

  • We start with the categroy $\mathsf{Sets}^{\mathrm{actv}}_*$ of pointed sets and — borrowing terminology from HA 2.1.2.1 — active morphisms of pointed sets, i.e. pointed maps $f\colon(X,x_0)\to(Y,y_0)$ satisfying $f^{-1}(y_0)=x_0$
  • Then, the free pointed set functor $(-)^+\colon\mathsf{Sets}\to\mathsf{Sets}^{\mathrm{actv}}_*$ can be shown to admit a left adjoint, given by the functor $(-)^-\colon\mathsf{Sets}^{\mathrm{actv}}_*\to\mathsf{Sets}$ sending $(X,x_0)$ to $X\setminus\{x_0\}$.
  • Taking categories of monoids with respect to the smash product and the Cartesian product gives a functor $$(-)^-\colon\mathsf{Mon}(\mathsf{Sets}^{\mathrm{actv}}_*)\to\mathsf{Mon}.$$ Here the objects in the former category are "integral monoids with zero", i.e. monoids $A$ equipped with a zero element $0$ satisfying $a0=a0=0$ for all $a\in A$ and such that $xy=0$ implies $x=0$ or $y=0$.
  • Finally, there is a group completion functor $\mathsf{Mon}\to\mathsf{Grp}$, so we obtain a composite $$\mathsf{Mon}(\mathsf{Sets}^{\mathrm{actv}}_*)\to\mathsf{Grp}.$$ The image of $(\mathbb{Z},\cdot,1)$ under this functor then recovers the multiplicative group of nonzero rational numbers $\mathbb{Q}^\times$.

Question. Can this procedure be adapted to the $\infty$-setting, and, assuming so, what would the group completion of $\mathbb{S}^-$, the "underlying multiplicative $\mathbb{E}_\infty$-monoid" of the sphere spectrum be?

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  • $\begingroup$ Looking at the background I wonder why you include commutativity in your question. $\endgroup$
    – Fernando Muro
    Commented Nov 11 at 19:46
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    $\begingroup$ One interpretation is that you group-complete the space of nonzero path components in $\Omega^\infty \Bbb S$, under the multiplicative product. I believe that if you do that, you get the $E_\infty$ space $\Bbb Q^\times$. The reasons should be the same as those here: mathoverflow.net/a/467856 $\endgroup$
    – Tyler Lawson
    Commented Nov 11 at 20:31
  • $\begingroup$ @FernandoMuro Just for motivation; there are analogous questions for the other $\mathbb{E}_n$-monoidal things, but I think the sphere spectrum is probably going to be the most important one. $\endgroup$
    – Emily
    Commented Nov 11 at 21:29
  • $\begingroup$ @TylerLawson Thanks! Do you think following the ∞-categorical version of the steps outlined for the 1-categorical case in the question will end up giving the space of nonzero path components in $\Omega^\infty\mathbb{S}$? $\endgroup$
    – Emily
    Commented Nov 11 at 21:44
  • $\begingroup$ (To be more precise, by "the $\infty$-categorical version of [...]" I have something like the following in mind: start with $\mathsf{Kan}^\mathrm{actv}_*$, take its homotopy coherent nerve to get an ∞-category $\mathcal{S}^\mathrm{actv}_*$ of spaces and active morphisms, show $(-)^+\colon\mathcal{S}\to\mathcal{S}^\mathrm{actv}_*$ admits a left adjoint $(-)^-$, check whether $\mathcal{S}$ satisfies the corresponding integrality condition (so that it is indeed an object of $\mathsf{Mon}_{\mathbb{E}_\infty}(\mathcal{S}^\mathrm{actv}_*)$), and then take $(-)^-)$ of $\mathbb{S}$.) $\endgroup$
    – Emily
    Commented Nov 11 at 21:46

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