The chain replacement of a poset flow

Abstract

We introduce the chain replacement of a poset flow: it is obtained by considering the simplicial nerves of the posets of strictly increasing chains in the given poset, ordered by refinement. It maps posets to q-cofibrant flows and inclusions of posets to q-cofibrations. Using the combinatorial properties of the chain replacement, we prove that pushouts along the chain replacement of an order-reflecting inclusion of posets induce trivial h-cofibrations on spaces of execution paths between states of the original flow. By introducing the Hurewicz model structure on flows (or H-model structure), we deduce the same property for any q-cofibrant replacement of an order-reflecting inclusion of posets.

arXiv comment28 pages; v2: finiteness hypothesis removed thanks to a final topology argument; many proofs were too compressed; v3: the Delta-Hausdorff case was not treated in Section 4