Model category structures in bifibred categories (Q1336806)

A bifibred category [\textit{A. Grothendieck}, ``Séminaire de géométrie algébrique 1'' (SGA 1), Lect. Notes Math. 224 (1971; Zbl 0234.14002)] can be seen as a particular case of a family of categories parametrized by objects of another category. For instance, the family of differential graded (dg) modules over a differential graded commutative (dgc) algebra is bifibred: the basis is the category of dgc algebras and fibers are the categories of dg modules over a specified dgc algebra. The main result of the work under review is the construction of a closed model category structure [\textit{D. G. Quillen}, ``Homotopical algebra'', Lect. Notes Math. 43 (1967; Zbl 0168.209)] on a bifibered category, from the same datum on basis and fibers (theorem 5.1). The example above is given in detail. As an application, the author shows that the Eilenberg-Moore spectral sequence of a fibration appears naturally in this setting, as a particular Quillen's derived functor.