Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

Question

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an input a symmetric monoidal groupoid S and produces a symmetric monoidal category S⁻¹S, whose objects are pairs (s,s') of objects in S and morphisms (s,s')→(t,t') are isomorphism classes of triples (A,A⊕s→t,A⊕s'→t').

Has this construction been investigated ∞-categorically, e.g., in the language of model categories?

I cannot even find a reference in the literature that shows that S⁻¹S represents (in the Thomason model structure) the homotopy group completion of S, so any references on this matter will be appreciated.

Once we do know that S↦S⁻¹S is the homotopy group completion functor, there is also the question of interpreting it as a left (derived) Quillen functor for some choice of a model structure on symmetric monoidal groupoids and (presumably) the Thomason model structure on categories.

A paper by Thomason (Beware the phony multiplication on Quillen's S⁻¹S) shows in particular that one cannot have an inverse functor x↦−x on S⁻¹S so that x⊕−x is naturally (strictly) isomorphic to 0. However, this does not preclude other ∞-categorical interpretations of the S⁻¹S construction.

Answer 1

A published reference for the claim (right after the question in boldface) is the proof given by Thomason on pages 1657-1658 of "First quadrant spectral sequences in algebraic K-theory via homotopy colimits", Comm. Algebra 10 (1982), no. 15, 1589–1668. He shows that if the translations are faithful, then S^{-1} S is equivalent to the mapping cone of the diagonal functor S --> S x S, and that this provides a group completion of S. Baas-Dundas-Richter-R. reviewed this in Lemma 6.2 of our paper on Ring completion of rig categories (Crelle, 2013), where we extend the story to the bimonoidal case.

Answer 2

Consider an $E_n$-monoid X. We can deloop $X$ to an $\infty$-category $\mathbf{B}X$. There's a natural functor $X^\circlearrowleft : \mathbf{B}X \rightarrow \text{Spc}$ given by the left action of $X$ on itself (essentially just the corepresented functor of the base point). Consider the unstraigthening of the functor $X^\circlearrowleft \times X^\circlearrowleft$, and call the corresponding left fibration $\mathbf{G} X \rightarrow \mathbf{B} X$. In recent work with Ramzi, we show that $| \mathbf{G} X | \cong X^{gp}$ provided that $n \geq 2$, i.e. the realization of $\mathbf{G} X$ computes the group completion.

The punchline is the following: If $S$ is a symmetric monoidal groupoid I can consider $S$ as an $E_\infty$-monoid. Then the homotopy category of the $\infty$-category $\mathbf{G} S$ is just Quillen's $S^{-1} S$. The assumption that Quillen introduces to show that $|S^{-1} S|$ actually computes the group completion, namely that $k+ : \text{Aut}(s) \rightarrow \text{Aut}(k+s)$ is injective for all $s, k \in S$, simply guarantees that $\mathbf{G} S$ is actually a $1$-category, i.e. equivalent to $S^{-1} S$.

A few remarks: There does exist an "inverse" functor $j : \mathbf{G} X \rightarrow \mathbf{G} X$, which becomes the inverse map after realization, and there do exist arrows $1 \rightarrow x \otimes j(x)$ for each $x \in \mathbf{G} X$. The problem that Thomason highlights is that these arrows are not part of a natural transformation.

Now, I'd argue that Quillen's $S^{-1}S$ should probably be replaced by the $\infty$-category $\mathbf{G} S$ for a general symmetric monoidal groupoid $S$. However, I'd still be doubtful that $\mathbf{G}$ becomes a natural left adjoint, since we haven't really considered the natural structure that $\mathbf{G} S$ has. To understand this, note that if we start with an $E_n$-monoid $X$, the $\infty$-category $\mathbf{G} X$ is only $E_{n-1}$-monoidal. So we have lost an $E_1$-structure along the way. (The same holds for Quillen's $S^{-1} S$. This problem is of course not immediately visible in the symmetric monoidal = $E_\infty$-case, but highlights one of the subtleties with the $S^{-1} S$ construction)

The extra structure is captured by the following: Consider the left fibration $\mathbf{E} X \rightarrow \mathbf{B} X$ which is the unstraightening of $X^\circlearrowleft$. Take the Cech nerve $G_\bullet(X)$ of the functor $\mathbf{E} X \rightarrow \mathbf{B} X$. This is automatically an internal groupoid in $E_{n-1}$-monoidal $\infty$-categories, with $G_1(X) = \mathbf{G}X$. We can also show that the realization of the simplicial object $G_\bullet(X)$ is equivalent to $\Omega B X$. I would be more hopeful to the claim that $G_\bullet(-)$ satisfies a universal property as a functor from $E_n$-monoids to internal groupoids in $E_{n-1}$-monoidal $\infty$-categories.