Some collapsing operations for 2-dimensional precubical sets

TL;DR

This paper introduces conditions for simplifying 2-dimensional precubical sets, which model concurrent systems, by collapsing edges or removing squares, thus aiding in constructing smaller, equivalent models in concurrency theory.

Contribution

It provides easily verifiable criteria for reducing 2D precubical sets while preserving their directed homotopy type, facilitating simpler models of concurrent systems.

Findings

Conditions for collapsing edges or squares are established.

Simplification preserves directed homotopy type.

Examples illustrate practical reduction methods.

Abstract

In this paper, we consider 2-dimensional precubical sets, which can be used to model systems of two concurrently executing processes. From the point of view of concurrency theory, two precubical sets can be considered equivalent if their geometric realizations have the same directed homotopy type relative to the extremal elements in the sense of P. Bubenik. We give easily verifiable conditions under which it is possible to reduce a 2-dimensional precubical set to an equivalent smaller one by collapsing an edge or eliminating a square and one or two free faces. We also look at some simple standard examples in order to illustrate how our results can be used to construct small models of 2-dimensional precubical sets.