  Our aim is to construct fibrewise 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
characterize completely those who admit a fibrewise nullification. As an
application we get fibrewise plus-construction and fibrewise Postnikov sections
for algebras over an operad.
