TITLE: $K$-theory backwards
QUESTION [12 upvotes]: Let $R$ be a ring. The $K$-theory of $(Mod(R)^{f.g.proj},\oplus)$ is obtained by first throwing out non-isomorphisms and then group completing. What happens if these steps are reversed?
That is, consider the symmetric monoidal $\infty$-category $C =
 (Mod(R)^{f.g.proj},\oplus)$ of finitely-generated projective $R$-modules under direct sum (of course, this is in fact an ordinary category). To compute $K(R)$, we first pass to the core $\iota(C)$ to get an $\infty$-groupoid (which in this case is a 1-groupoid) and then we group complete to get $K(R) = \iota(C)[C^{-1}]$.
I'm thinking of group completion as an $\infty$-categorical process: for a symmetric monoidal $\infty$-category $C$, the group completion $C[C^{-1}]$ is the free symmetric monoidal $\infty$-category on $C$ where every object has a monoidal inverse; this agrees with the usual notion for symmetric monoidal $\infty$-groupoids. So it makes sense to reverse these steps and consider $\iota(C[C^{-1}])$. So my question is:
Question:

If $C$ is a symmetric monoidal $\infty$-category, then do we have $\iota(C)[C^{-1}] = \iota(C[C^{-1}])$?
If not, does this at least hold in the case $C = Mod(R)^{f.g.proj}$, yielding an alternate description of $K(R)$?

Motivation: My motivation here is basically to explore how robust the definition of $K$-theory is to perturbation. For example, an alternate description of $K(R)$ that definitely does work is the following: $K(R) = \iota_\oplus(\iota(Mod(R))[R^{-1}])$: that is, take the groupoid of all $R$-modules and rather than group completing, monoidally invert the single object $R$ itself viewed as an $R$-module; finally, throw away all objects which are not monoidally invertible (note in particular that since we're not monoidally inverting everything, we've sidestepped the Eilenberg swindle). I like this description because you don't have to put the "finitely-generated projective" condition in by hand -- it falls out naturally just from knowing that that the symmetric monoidal category $(Mod(R),\oplus)$ has a distinguished object $R$. If the answer to my question is "yes", then one could simplify this further to $K(R) \overset{?}{=} \iota_\oplus(\iota(Mod(R)[R^{-1}]))$; then even the choice of the $\oplus$ monoidal structure becomes more "canonical" in the sense that it has a universal property in $Mod(R)$, unlike in $\iota(Mod(R))$. Other variants start to suggest themselves, too.

REPLY [3 votes]: The only way I know to speak of `group-completion' of a symmetric monoidal $\infty$-category involves not only adjoining inverses of all objects but also inverses of all morphisms. In that sense, group-completion corresponds to first taking the classifying space (rather than the core) and then group-completing. For example, if $\mathcal{C}$ has an initial or terminal object, its classifying space is trivial, so its group completion is trivial.
Of course there are plenty of decent symmetric monoidal categories where every object has an inverse but which are not groupoids (for example, the full subcategory of $\text{Mod}_R$ on invertible modules). But maybe this does not deserve to be called `group-complete' because the inverse is not functorial (at least, not covariantly).
If there is a functorial inverse $-\text{id}:\mathcal{C}\rightarrow\mathcal{C}$ such that $-\text{id}\otimes\text{id}$ is the zero functor, then the category is a groupoid. Specifically, each morphism $f:X\rightarrow Y$ has an inverse (up to equivalence) $X\otimes Y\otimes(-f):Y\rightarrow X$.
EDIT: Why would we want the inverse to be covariantly functorial? Let's say for example that $\mathcal{R}$ is a semiring $\infty$-category which has additive inverses. Then multiplication by any element (in particular -1) is functorial $\mathcal{R}\rightarrow\mathcal{R}$, so $\mathcal{R}$ is an $\infty$-groupoid as above.
Thus group-complete symmetric monoidal $\infty$-categories are exactly grouplike symmetric monoidal $\infty$-groupoids, also known as grouplike $\mathbb{E}_\infty$-spaces or connective spectra. The universal example of one of these is the sphere spectrum $\mathbb{S}$. Modules over $\mathbb{S}$ (in the sense of modules over a semiring $\infty$-category) are precisely the connective spectra, and the group-completion operation you want is $$-\otimes\mathbb{S}:\text{SymMon}\rightarrow\text{Mod}_\mathbb{S}\cong\text{Sp}_{\geq 0},$$ analogous to $$-\otimes\mathbb{Z}:\text{ComMon}\rightarrow\text{Mod}_\mathbb{Z}\cong\text{Ab}.$$ I discuss all of this in section 4 of my paper https://arxiv.org/abs/1606.05606