Combinatorics of labelling in higher dimensional automata

TL;DR

This paper explores the combinatorics of labelling in higher dimensional automata, introducing a new construction called labelled transverse symmetric precubical sets to ensure well-behaved labelling functors, and demonstrating their equivalence to existing models.

Contribution

It introduces labelled transverse symmetric precubical sets and proves their uniqueness and equivalence to traditional models, improving the mathematical framework for concurrent process labelling.

Findings

The labelled coskeleton functor can be made well-behaved using transverse symmetric precubical sets.

The proposed solution is unique among possible constructions.

A new semantics of CCS is derived, equivalent to the existing one.

Abstract

The main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled n-cube, in exactly one way. The main ingredient is the non-functorial construction called labelled directed coskeleton. It is defined as a subobject of the labelled coskeleton, the latter coinciding in the unlabelled case with the right adjoint to the truncation functor. This non-functorial construction is necessary since the labelled coskeleton functor of the category of labelled precubical sets does not fulfil the above requirement. We prove in this paper that it is possible to force the labelled coskeleton functor to be well-behaved by working with labelled transverse symmetric precubical sets. Moreover, we prove that this solution is the only one. A transverse symmetric precubical set is a precubical set equipped with symmetry maps and with a new kind of degeneracy map called transverse degeneracy. Finally, we also prove that the two settings are equivalent from a directed algebraic topological viewpoint. To illustrate, a new semantics of CCS, equivalent to the old one, is given.