Strong homotopy types of acyclic categories and $\mathrm{\Delta }$-complexes

We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and $\mathrm{\Delta }$-complexes, respectively. The functors of classifying spaces and face posets are compatible with these homotopy theories. In contrast with the classical settings of finite spaces and simplicial complexes, the universality of morphisms and simplices plays a central role in this paper.