Homotopical equivalence of combinatorial and categorical semantics of process algebra

TL;DR

This paper demonstrates that combinatorial and categorical semantics of CCS are homotopically equivalent, simplifying the semantics by working directly in the homotopy category of flows without losing geometric information.

Contribution

It establishes a homotopical equivalence between combinatorial and categorical semantics of CCS, enabling a purely homotopical approach to process algebra semantics.

Findings

Combinatorial semantics can be translated into categorical semantics via a geometric realization.

Homotopy equivalence preserves geometric information of the semantics.

Categorical semantics simplifies the combinatorial complexity of process algebra.

Abstract

It is possible to translate a modified version of K. Worytkiewicz's combinatorial semantics of CCS (Milner's Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires non-canonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomorphic if and only if the associated flows are weakly equivalent. The interest of the categorical semantics is that combinatorics totally disappears. Last but not least, a part of the categorical semantics of CCS goes down to a pure homotopical semantics of CCS using A. Heller's privileged weak limits and colimits. These results can be easily adapted to any other process algebra for any synchronization algebra.