Lifting of model structures to fibred categories

A fibred category 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 fibred category $p: \mathbf{N} \to \mathbf{M}$ and a model structure on the "base 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 category $\mathbf{M}$ may be lifted to right homotopic maps in the fibred 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 $\mathbf{N}$.

Keywords

fibred category, model category

2010 Mathematics Subject Classification

55U35

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Published 25 January 2012