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.