Generalized Reedy diagrams in tribes

TL;DR

This paper develops a new framework for constructing Reedy diagram categories in tribes, enabling the establishment of tribe structures on fibrant diagrams for generalized inverse categories.

Contribution

It introduces a method to build Reedy categories from generalized Reedy categories and applies this to define tribe structures on fibrant diagrams in tribes.

Findings

Constructed an absolutely dense functor from a strict Reedy category to a generalized Reedy category.

Provided a tribe structure on a subcategory of fibrant diagrams in a given tribe.

Extended Reedy diagram theory to generalized inverse categories.

Abstract

Starting from a generalized Reedy category $R$ satisfying a simple condition, we construct an absolutely dense functor $\mathbf{D}_R \to R$ with domain a strict Reedy category. In the case of a generalized inverse category $R$, and given any tribe $\mathcal{T}$, we leverage this construction to provide a tribe structure on a subcategory of fibrant diagrams in $\mathcal{T}^R$.