Comparison Between Different Topological Models of Concurrency

TL;DR

This paper compares various topological models of concurrency, establishing equivalences between them and introducing new categorical frameworks to better understand concurrent computing.

Contribution

It provides explicit non-Quillen and Quillen equivalences between key models of concurrency and introduces the category of boxed symmetric trees as a flexible new framework.

Findings

Explicit non-Quillen equivalence between precubical sets and flows.

Quillen equivalence between simplicial semicategories and flows.

Boxed symmetric trees form a test category with potential for modeling concurrency.

Abstract

In this note, we provide an explicit non-Quillen equivalence between the category of precubical sets and Gaucher's category of flows via a class of "realization functors" (with mild assumptions on the cofibrations of the category of precubical sets). In addition, we demonstrate a Quillen equivalence between simplicial semicategories and flows before proving that simplicial semicategories satisfy many of the same properties as flows. Finally, we introduce the category of boxed symmetric trees, presheaves on which may provide a slightly more flexible setting for concurrent computing than (pre)cubical sets, before showing that when endowed with degeneracies, the aforementioned presheaf category is a test category (although not strict test).