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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.

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  • $\begingroup$ do you want to see $S\mapsto S^{-1}S$ as left adjoint or as homotopy left adjoint ? Would you be satisfied if you translate every thing in terms of $E_{\infty}$-spaces (since you have mentioned Thomason model structure). $\endgroup$
    – Ilias A.
    Nov 30, 2015 at 10:37
  • $\begingroup$ @IliasAmrani: I might be wrong, but I think the implicit folklore understanding is that S⁻¹S computes the homotopy left adjoint. I'll add “derived” to the post. I will also be happy with an interpretation using group-like E_∞-spaces of any kind. $\endgroup$
    – Dmitri Pavlov
    Nov 30, 2015 at 15:15
  • $\begingroup$ @IliasAmrani: I guess I misunderstood the intended meaning of “translate … E_∞-spaces”: showing that some other well-known formula for the homotopy group completion (e.g., ΩB) implements the homotopy group completion functor is not a part of the question. I am interested exclusively in S⁻¹S, not some other construction. $\endgroup$
    – Dmitri Pavlov
    Nov 30, 2015 at 20:36
  • $\begingroup$ No problem, that happens :). You wrote "I cannot $\mathbf{even}$ 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." That is why i concluded that you were asking also if this functor is homotopy left adjoint under hypothesis that you know that it represent complition... $\endgroup$
    – Ilias A.
    Nov 30, 2015 at 20:44
  • $\begingroup$ @IliasAmrani: I now see where the misunderstanding comes from. My intended meaning of “even” was that even if one does know that S⁻¹S is the homotopy group completion of S, there is still the question of which model structures to choose on both sides so that S⁻¹S preserves (acyclic) cofibrations (and hence is a left Quillen functor). $\endgroup$
    – Dmitri Pavlov
    Nov 30, 2015 at 20:50

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