Lifting of model structures to fibred categories

Abstract

A consists of a functor $p: \mathbf{N} \to \mathbf{M}$ between categories $\mathbf{N}$ and $\mathbf{M}$ such that objects of $\mathbf{N}$ may be pulled back along any arrow of $\mathbf{M}$. Given a $p: \mathbf{N} \to \mathbf{M}$ and a model structure on the category $\mathbf{M}$, we show that there exists a lifting of the model structure on $\mathbf{M}$ to a model structure on $\mathbf{N}$. We will refer to such a system as a fibred model category and give several examples of such structures. We show that, under certain conditions, right homotopies of maps in the base $\mathbf{M}$ may be lifted to right homotopic maps in the category. Further, we show that these lifted model structures are well behaved with respect to Quillen adjunctions and Quillen equivalences. Finally, we show that if $\mathbf{N}$ and $\mathbf{M}$ carry compatible closed monoidal structures and the functor $p$ commutes with colimits, then a Quillen pair on $\mathbf{M}$ lifts to a Quillen pair on $\mathbf{N}$.