Functors between Reedy model categories of diagrams

TL;DR

This paper characterizes Reedy functors that induce Quillen adjunctions between diagram categories, with applications to fibrant subdiagrams of multicosimplicial objects, advancing the understanding of model structures on diagram categories.

Contribution

It provides a characterization of Reedy functors that induce Quillen functors between diagram categories for all model categories, extending the theory of Reedy model structures.

Findings

Identifies conditions for Reedy functors to induce Quillen functors

Shows certain subdiagrams of fibrant multicosimplicial objects are fibrant

Enhances understanding of model structures on diagram categories

Abstract

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