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Homological localisation of model categories. (English) Zbl 1318.55009
A stable model category is a model category of a particular kind; in particular, if \({\mathcal C}\) is a stable model category then the homotopy category \(\text{Ho}({\mathcal C})\) is a module over the stable homotopy category \(\text{Ho}({\mathcal S})\).
Let \(E\) be a spectrum, and consider \(\text{Ho}(L_E{\mathcal S})\), the Bousfield localisation of the stable homotopy category at \(E\). Given a well-behaved stable model category \({\mathcal C}\), the authors construct a new model structure \({\mathcal C}_E\) on \({\mathcal C}\) such that the action of \(\text{Ho}({\mathcal S})\) on \(\text{Ho}({\mathcal C}_E)\) factors through \(\text{Ho}(L_E{\mathcal S})\), and they show that \({\mathcal C}_E\) has a universal property. Also, for a stable model category \({\mathcal C}\) with an equivalence \(\Phi: \text{Ho}(L_E{\mathcal S})\to \text{Ho}({\mathcal C})\), they show that \(L_E{\mathcal S}\) and \({\mathcal C}\) are Quillen equivalent if and only if \(\Phi\) is a equivalence of \(\text{Ho}({\mathcal S})\)-module categories.

MSC:
55P42 Stable homotopy theory, spectra
55P60 Localization and completion in homotopy theory
18E30 Derived categories, triangulated categories (MSC2010)
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