Functors between Reedy model categories of diagrams

Abstract

If \(\mathcal{D}\) is a Reedy category and \(\mathcal{M}\) is a model category, the category \(\mathcal{M}^{\mathcal{D}}\) of \(\mathcal{D}\)-diagrams in \(\mathcal{M}\) is a model category under the Reedy model category structure. If \(\mathcal{C} \to \mathcal{D}\) is a Reedy functor between Reedy categories, then there is an induced functor of diagram categories \(\mathcal{M}^{\mathcal{D}} \to \mathcal{M}^{\mathcal{C}}\). Our main result is a characterization of the Reedy functors \(\mathcal{C} \to \mathcal{D}\) that induce right or left Quillen functors \(\mathcal{M}^{\mathcal{D}} \to \mathcal{M}^{\mathcal{C}}\) for every model category \(\mathcal{M}\). We apply these results to various situations, and in particular show that certain important subdiagrams of a fibrant multicosimplicial object are fibrant.