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 $G$.
Categories: math.AT math.CT