There is a natural notion of a subdivision of a lower Eulerian poset called a strong formal subdivision, which abstracts the notion of a polyhedral subdivision of a polytope, or a proper, surjective morphism of fans. We show that there is a canonical bijection between strong formal subdivisions and triples consisting of a lower Eulerian poset, a corresponding rank function, and a non-minimal element such that the join with any other element exists. The bijection uses the non-Hausdorff mapping cylinder construction introduced by Barmak and Minian. A corresponding bijection for $CW$-posets is given, as well as an application to computing the $cd$-index of an Eulerian poset. A companion paper explores applications to Kazhdan-Lusztig-Stanley theory.