Globular realization and cubical underlying homotopy type of time flow of process algebra

TL;DR

This paper introduces a small, efficient realization method for precubical sets modeling process algebra, establishing a connection between globular and cubical homotopy types without complex constructions.

Contribution

It provides a novel, minimal realization construction for precubical sets and demonstrates the equivalence of globular and cubical homotopy types for these models.

Findings

Realization of precubical sets as flows without cofibrant replacements

Finite precubical sets lead to flows with finite globular decompositions

Underlying homotopy type matches the standard cubical complex homotopy type

Abstract

We construct a small realization as flow of every precubical set (modeling for example a process algebra). The realization is small in the sense that the construction does not make use of any cofibrant replacement functor and of any transfinite construction. In particular, if the precubical set is finite, then the corresponding flow has a finite globular decomposition. Two applications are given. The first one presents a realization functor from precubical sets to globular complexes which is characterized up to a natural S-homotopy. The second one proves that, for such flows, the underlying homotopy type is naturally isomorphic to the homotopy type of the standard cubical complex associated with the precubical set.