Local framings.

*(English)*Zbl 1260.55010A stable model category \(\mathcal{C}\) is called “rigid” if its homotopy category \(\mathrm{Ho}(\mathcal{C})\) depends (in a precise homotopy-theoretic sense) only on the triangulated structure of \(\mathrm{Ho}(\mathcal{C})\). The main (outstanding) work on rigid stable model categories is the paper [S. Schwede, Ann. Math. (2) 166, No. 3, 837–863 (2007; Zbl 1151.55007)], and there is more progress along these lines by the second author of the paper under review in [C. Roitzheim, Geom. Topol. 11, 1855–1886 (2007; Zbl 1142.55007)]. This last work concerns a localization of the category of spectra and, in the paper under review, the authors develop some machinery to help with further work on the rigidity question in localized settings.

In more detail, in the local setting for an arbitrary homology theory \(E_\ast\), there is the following rigidity question: if \(\mathcal{C}\) is a stable model category such that there is an equivalence of triangulated categories between \(\mathrm{Ho}(L_E\mathcal{S})\) and \(\mathrm{Ho}(\mathcal{C})\), where \(\mathcal{S}\) is the model category of Bousfield-Friedlander spectra, then does it follow that \(L_E\mathcal{S}\) and \(\mathcal{C}\) are Quillen equivalent?

To help with making progress on the above question, the authors prove that if \(\mathcal{C}\) is a stable model category, then the action of \(\mathrm{Ho}(\mathcal{S})\) on \(\mathrm{Ho}(\mathcal{C})\) yields an action of \(\mathrm{Ho}(L_E\mathcal{S})\) if and only if the derived mapping spectrum \(\mathrm{Map}(X,Y)\) is \(E\)-local for all \(X\) and \(Y\) in \(\mathcal{C}\). When this condition is satisfied, then \(\mathcal{C}\) is referred to as “stably \(E\)-familiar”; this notion is developed after considering a similar notion in the setting of simplicial sets.

Thanks to the above result and others, the authors show that if \(L_E\) is a smashing localization, then there is a Quillen equivalence between \(\mathcal{C}\) and \(L_E\mathcal{S}\) if and only if (a) \(\mathcal{C}\) is stably \(E\)-familiar, and (b) the map \(\mathbb{S} \to R\text{Map}(X,X)\) is an \(E\)-equivalence, whenever \(X\) is a fibrant-cofibrant compact generator of \(\mathcal{C}\). If properties (a) and (b) depend only on the triangulated structure of \(\mathrm{Ho}(\mathcal{C})\), then \(\mathrm{Ho}(L_E\mathcal{S})\) is rigid. This paper also contains other interesting results.

In more detail, in the local setting for an arbitrary homology theory \(E_\ast\), there is the following rigidity question: if \(\mathcal{C}\) is a stable model category such that there is an equivalence of triangulated categories between \(\mathrm{Ho}(L_E\mathcal{S})\) and \(\mathrm{Ho}(\mathcal{C})\), where \(\mathcal{S}\) is the model category of Bousfield-Friedlander spectra, then does it follow that \(L_E\mathcal{S}\) and \(\mathcal{C}\) are Quillen equivalent?

To help with making progress on the above question, the authors prove that if \(\mathcal{C}\) is a stable model category, then the action of \(\mathrm{Ho}(\mathcal{S})\) on \(\mathrm{Ho}(\mathcal{C})\) yields an action of \(\mathrm{Ho}(L_E\mathcal{S})\) if and only if the derived mapping spectrum \(\mathrm{Map}(X,Y)\) is \(E\)-local for all \(X\) and \(Y\) in \(\mathcal{C}\). When this condition is satisfied, then \(\mathcal{C}\) is referred to as “stably \(E\)-familiar”; this notion is developed after considering a similar notion in the setting of simplicial sets.

Thanks to the above result and others, the authors show that if \(L_E\) is a smashing localization, then there is a Quillen equivalence between \(\mathcal{C}\) and \(L_E\mathcal{S}\) if and only if (a) \(\mathcal{C}\) is stably \(E\)-familiar, and (b) the map \(\mathbb{S} \to R\text{Map}(X,X)\) is an \(E\)-equivalence, whenever \(X\) is a fibrant-cofibrant compact generator of \(\mathcal{C}\). If properties (a) and (b) depend only on the triangulated structure of \(\mathrm{Ho}(\mathcal{C})\), then \(\mathrm{Ho}(L_E\mathcal{S})\) is rigid. This paper also contains other interesting results.

Reviewer: Daniel G. Davis (Lafayette)