A case of monoidal uniqueness of algebraic models (Q889600)

\textit{S. Schwede}'s rigidity theorem [Ann. Math. (2) 166, No. 3, 837--863 (2007; Zbl 1151.55007)] asserts that the stable homotopy category has a unique model: If there is a triangulated equivalence \(\mathrm{Ho}(\mathrm{Sp}) \to \mathrm{Ho}(\mathcal{C})\) for some stable model category \(\mathcal{C}\), then \(\mathcal{C}\) is Quillen equivalent to the category \(\mathrm{Sp}\) of spectra. Rigidity has also been studied for modules over ring spectra other than the sphere spectrum \(S\), and for various localizations of the stable homotopy category. A related problem is the search for simpler models, for instance, algebraic models. A model category is called \textit{algebraic}, or a \(\mathrm{Ch}\)-model category, if it is enriched, tensored, and cotensored in chain complexes. The paper under review investigates algebraic models for modules over the \(K(1)\)-local sphere \(L_{K(1)}S\) at odd primes. The main result is that the category of \(L_{K(1)}S\)-modules admits \textit{at most} one algebraic model. More precisely, \textit{if} there is a unit-preserving triangulated equivalence \[ \mathrm{Ho} \left( (L_{K(1)}S)\mathrm{-Mod} \right) \to \mathrm{Ho}(\mathcal{C}) \] for some monoidal algebraic model category \(\mathcal{C}\), then \(\mathcal{C}\) is Quillen equivalent to the category of dg-modules over some explicit differential graded algebra (dga). Building on work of Schwede--Shipley [\textit{S. Schwede} and \textit{B. Shipley}, Topology 42, No.1, 103--153 (2003; Zbl 1013.55005)] and Dugger--Shipley [\textit{D. Dugger} and \textit{B. Shipley}, Adv. Math. 212, No. 1, 37--61 (2007; Zbl 1118.55008)], the proof reduces to a statement about dga's. The author constructs an explicit commutative dga \(C\) whose homology \(H_*C\), equipped with 3-fold Massey products, is isomorphic to the homotopy groups \(\pi_* L_{K(1)}S\), equipped with 3-fold Toda brackets. She then proves that if \(D\) is any commutative dga with the same homology and 3-fold Massey products, then there is a quasi-isomorphism of dga's \(C \to D\).