Directed degeneracy maps for precubical sets

TL;DR

This paper develops directed homotopy theory for symmetric transverse sets, introducing realization functors to flows and establishing their homotopy equivalence, thus generalizing prior results from precubical sets.

Contribution

It introduces a q-realization functor and a natural realization functor for symmetric transverse sets, proving their homotopy equivalence and extending previous work on precubical sets.

Findings

Construction of cotransverse topological cube as a cotransverse Lawvere metric space

Homotopy equivalence of realization functors on cofibrant symmetric transverse sets

Generalization of results from precubical sets to symmetric transverse sets

Abstract

Symmetric transverse sets were introduced to make the construction of the parallel product with synchronization for process algebras functorial. It is proved that one can do directed homotopy on symmetric transverse sets in the following sense. A q-realization functor from symmetric transverse sets to flows is introduced using a q-cofibrant replacement functor of flows. By topologizing the cotransverse maps, the cotransverse topological cube is constructed. It can be regarded both as a cotransverse topological space and as a cotransverse Lawvere metric space. A natural realization functor from symmetric transverse sets to flows is introduced using Raussen's notion of natural $d$-path extended to symmetric transverse sets thanks to their structure of Lawvere metric space. It is proved that these two realization functors are homotopy equivalent on cofibrant symmetric transverse sets by using the fact that the small category defining symmetric transverse sets is c-Reedy in Shulman's sense. This generalizes to symmetric transverse sets results previously obtained for precubical sets.