Lifting of model structures to fibred categories (Q719058)

Fibred categories where introduced in [\textit{A. Grothendieck et al.}, Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1), dirigé par Alexander Grothendieck. Augmenté de deux exposés de M. Raynaud. Lecture Notes in Mathematics 224. Berlin-Heidelberg-New York: Springer-Verlag. (1971; Zbl 0234.14002)]. A fibred category consists of a functor \(p:N\to M\) between categories \(N\) and \(M\) such that objects of \(N\) may be ``pulled back along any arrow of \(M\)''. The author describes in this paper a process whereby a model structure may be lifted from a base category \(M\) to a category \(N\) fibered over \(M\) via the functor \(p:N\to M\). Several examples of this lifted model structures are given and it is shown that the lifted model structures are well behaved with respect to Quillen adjunctions and Quillen equivalences on the base category. Moreover, it is shown that if \(N\) and \(M\) carry compatible closed monoidal structures and the functor \(p\) commutes with colimits, then a Quillen pair on \(M\) lifts to a Quillen pair on \(N\).