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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)
##### Keywords:
stable model category; Bousfield localisation
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##### References:
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