Generalized Reedy diagrams in tribes
El Mehdi Cherradi
TL;DR
The work generalizes Reedy diagram theory to generalized Reedy categories by unrolling $R$ into a strict Reedy $oldsymbol{D}_{R}$ and proving $p:oldsymbol{D}_{R}\to R$ is absolutely dense. For generalized inverse $R$ and a tribe $oldsymbol{T}$, it constructs a tribe of $p$-fibrant diagrams in $oldsymbol{T}^{R^{op}}$ by using precomposition along fibering functors and the right Kan extension $p_*$, preserving fibrations and anodyne maps and supporting internal products. The approach yields a robust framework for handling diagrams in tribes beyond inverse categories, with concrete examples (e.g., group actions) illustrating the behavior and highlighting distinctions from earlier fibrational models. Overall, the paper provides a modular method to transfer Reedy-type fibrancy and tribe structures along a dense projection, enabling new fibrant diagram categories and pi-tribe stability.
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 $\mathcal{T}^R$.