In this paper we explain when it is possible to construct fiberwise localizations in model categories. For pointed spaces, the general idea is to decompose the total space of a fibration as a diagram over the category of simplices of the base and replace it by the localized diagram. This of course is not possible in an arbitrary category. We have thus to adapt another construction which heavily depends on Mather's cube theorem. Working with model categories in which the cube theorem holds, we propose a few equivalent conditions under which fiberwise nullifications exist. We show that these techniques apply to yield a fiberwise plus-construction for differential graded algebras over cofibrant operads.
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David Chataur, Jérôme Scherer, Fiberwise localization and the cube theorem. Comment. Math. Helv. 81 (2006), no. 1, pp. 171–189DOI 10.4171/CMH/48