Group completion and units in \(\mathcal I\)-spaces
Let \(\mathcal{I}\) be the category whose objects are the empty set and finite sets of the form \(\{1, 2, ..., n\}\), with morphisms the injective functions. Then the category \(\mathcal{S}^\mathcal{I}\) of \(\mathcal{I}\)-spaces is the symmetric monoidal category of functors \(\mathcal{I} \to \mathcal{S}\), where \(\mathcal{S}\) is the category of simplicial sets and the monoidal structure is induced by concatenation in \(\mathcal{I}\). A commutative monoid in \(\mathcal{S}^\mathcal{I}\) is known as a commutative \(\mathcal{I}\)-space monoid and, following the authors of this paper, we use \(\mathcal{C}\mathcal{S}^\mathcal{I}\) to denote the category of commutative monoids in \(\mathcal{S}^\mathcal{I}\). In [\textit{S. Sagave} and \textit{C. Schlichtkrull}, Adv. Math. 231, No. 3--4, 2116--2193 (2012; Zbl 1315.55007)], the authors of the paper under review show that \(\mathcal{C}\mathcal{S}^\mathcal{I}\) can be equipped with the ``positive \(\mathcal{I}\)-model structure,'' and the resulting model category is Quillen equivalent to the category of \(E_\infty\)-spaces. Also, if \(A \in \mathcal{C}\mathcal{S}^\mathcal{I}\), then the associated space \(A_{h\mathcal{I}}\), the homotopy colimit of the diagram \(A\), is an \(E_\infty\)-space. We now describe a result from the paper under review. Let \(B \: \mathcal{C}\mathcal{S}^\mathcal{I} \to \mathcal{C}\mathcal{S}^\mathcal{I}\) be the bar construction with right adjoint the loop functor \(\Omega\), and, with \(A\) as above, let \(B(A) \to B(A)^{\mathcal{I}-\mathrm{fib}}\) be a functorial fibrant replacement in the positive \(\mathcal{I}\)-model structure on \(\mathcal{C}\mathcal{S}^\mathcal{I}\). The authors show that if \(A\) is cofibrant in \(\mathcal{C}\mathcal{S}^\mathcal{I}\), then the natural map \[ A_{h\mathcal{I}} \to \bigl(\Omega(B(A)^{\mathcal{I}-\mathrm{fib}})\bigr)_{h\mathcal{I}} \] is a group completion. Additionally, there is a version of this result that shows that group completion for topological monoids via the bar construction lifts to \(\mathcal{C}\mathcal{S}^\mathcal{I}\). It is shown in this paper that \(\mathcal{C}\mathcal{S}^\mathcal{I}\) has a model structure in which group completion can be obtained as a fibrant replacement. This may be the first time group completion in the \(E_\infty\) setting has been given by fibrant replacement. Following the authors, let \(\mathcal{C}\mathcal{S}^\mathcal{I}_{\mathrm{gp}}\) denote the category \(\mathcal{C}\mathcal{S}^\mathcal{I}\) equipped with this ``group completion model structure.'' Then the authors show that there is a Quillen equivalence between \(\mathcal{C}\mathcal{S}^\mathcal{I}_{\mathrm{gp}}\) and the model category of \(\Gamma\)-spaces (equipped with the stable \(Q\)-model structure). A corollary of this result is that the homotopy category of grouplike commutative \(\mathcal{I}\)-space monoids is equivalent to the homotopy category of connective spectra. Just as an \(E_\infty\)-space \(X\) has a subspace \(X^\times\) of units, for any \(A \in \mathcal{C}\mathcal{S}^\mathcal{I}\), there is a submonoid \(A^\times\) of units which satisfies the desired ``consistency property'' \[ (A^\times)_{h\mathcal{I}} \cong (A_{h\mathcal{I}})^\times. \] The construction of \(A^\times\) has proven useful: for example, it plays a role in [\textit{A. J. Blumberg, R. L. Cohen} and \textit{C. Schlichtkrull}, Geom. Topol. 14, No. 2, 1165--1242 (2010; Zbl 1219.19006)]. In the paper under review, \(\mathcal{C}\mathcal{S}^\mathcal{I}\) is shown to have a ``units model structure'' in which the formation of units can be viewed as a cofibrant replacement. Additionally, the authors give several other interesting results that involve the units model structure.
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