Reedy categories and the $\Theta$-construction

TL;DR

This paper demonstrates that the $ heta$-construction preserves Reedy categories, introduces elegant Reedy categories, and shows their Reedy and injective model structures coincide, providing new insights into categorical structures.

Contribution

It proves that the $ heta$-construction preserves Reedy categories and introduces elegant Reedy categories with coinciding Reedy and injective model structures.

Findings

$ heta ext{C}$ is Reedy if $ ext{C}$ is Reedy

Categories $ heta_n$ are Reedy

Reedy and injective model structures coincide for elegant Reedy categories

Abstract

We use the notion of multi-Reedy category to prove that, if $\mathcal C$ is a Reedy category, then $\Theta \mathcal C$ is also a Reedy category. This result gives a new proof that the categories $\Theta_n$ are Reedy categories. We then define elegant Reedy categories, for which we prove that the Reedy and injective model structures coincide.