Title: Strong homotopy types of acyclic categories and $Δ$-complexes,
Abstract: We extend the homotopy theories based on point reduction for finite spaces
and simplicial complexes to finite acyclic categories and $\Delta$-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.