Generalized equivariant model structures on $\mathbf{Cat}^I$

TL;DR

This paper develops a generalized framework for equivariant model structures on categories of small categories, acyclic categories, and posets, establishing their Quillen equivalences with simplicial sets under certain conditions.

Contribution

It introduces a broad class of equivariant model structures on $ extbf{Cat}^I$, extending previous group-based results to more general small categories and object classes.

Findings

Established $ ext{O}$-equivariant model structures on $ extbf{Cat}^I$, $ extbf{Ac}^I$, and $ extbf{Pos}^I$

Proved Quillen equivalences with $ extbf{sSet}^I$ model structures

Generalized previous results from group actions to arbitrary small categories

Abstract

Let $I$ be a small category, $\mathcal{C}$ be the category $\mathbf{Cat}$, $\mathbf{Ac}$ or $\mathbf{Pos}$ of small categories, acyclic categories, or posets, respectively. Let $\mathcal{O}$ be a locally small class of objects in $\mathbf{Set}^I$ such that $\mathrm{colim}_I O=*$ for every $O\in \mathcal{O}$. We prove that $\mathcal{C}^I$ admits the $\mathcal{O}$-equivariant model structure in the sense of Farjoun, and that it is Quillen equivalent to the $\mathcal{O}$-equivariant model structure on $\mathbf{sSet}^I$. This generalizes previous results of Bohmann-Mazur-Osorno-Ozornova-Ponto-Yarnall and of May-Stephan-Zakharevich when $I=G$ is a discrete group and $\mathcal{O}$ is the set of orbits of $G$.