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

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 ${\mathrm{colim}}_{I}O=\ast$ 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 $G$.