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The Grayson—Quillen 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^{−1}S, 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?
For example, can we interpret S↦S^{−1}S as a left Quillen functor for some model structure on symmetric monoidal groupoids and the Thomason model structure on categories?
In fact, I cannot even find a reference in the literature that shows that S^{−1}S represents (in the Thomason model structure) the homotopy group completion of S.
Weirdly enough, nLab doesn’t seem to have an article on the Grayson—Quillen construction. Should I create one, or perhaps it’s buried in some other article?
One problem might be that the category S^{-1}S does not have an obvious universal property. This is at the heart of the famous mistake that was pointed out by Thomason in his paper “Beware the phony multiplication on Quillen’s S^{-1}S”. There is an obvious universal property that it looks like it should have. There are two functors f,g from S to S^{-1}S (given by including in the two obvious ways). These should be negatives of each other. And it looks like S^{-1}S should be the universal thing such that there is a natural transformation from 0 to f + g. There is even a candidate for the natural transformation. However there is a problem. This candidate natural transformation is not actually natural, and there is not a different choice making it natural.
@Chris: Yes, Thomason’s remarks indeed rule out the possibility that objects in the target category have a map x→x^{−1} such that the relevant identities for inverses are witnessed by (strict!) natural transformations.
But this still leaves open the question whether we can relax this notion of inverse in the target category (e.g., by using some kind of a homotopy coherent natural transformation) so that the above problem is resolved.
Okay.
There is a Quillen Q-construction and a Quillen plus construction, so I’m wondering about the best way to distinguish this construction from the other two. Are there any standard names for this in the literature?
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