Model category structures in bifibred categories

Abstract

In many situations we encounter categories which are, in some sense, parametrized by objects belonging to other categories. Given such a category with model category structures in the parametrizing category and in the fibre categories, the main result of this paper is to show how to endow the total category with a model category structure. As a major application, we show that, in this setting, a particular Quillen derived functor gives rise to the differential torsion product, relevant to the Eilenberg-Moore spectral sequence of a fibration.