The choice of cofibrations of higher dimensional transition systems

TL;DR

This paper investigates model structures on higher dimensional transition systems, revealing that fibrant replacements often disrupt causal structures, and proposes star-shaped systems as a solution.

Contribution

It establishes a left determined model structure for weak transition systems and analyzes its implications on fibrant replacements and causal structures.

Findings

Fibrant replacements often contain transitions between all state pairs.

Model structures restrict to cubical and regular transition systems.

Star-shaped transition systems preserve causal structures.

Abstract

It is proved that there exists a left determined model structure of weak transition systems with respect to the class of monomorphisms and that it restricts to left determined model structures on cubical and regular transition systems. Then it is proved that, in these three model structures, for any higher dimensional transition system containing at least one transition, the fibrant replacement contains a transition between each pair of states. This means that the fibrant replacement functor does not preserve the causal structure. As a conclusion, we explain why working with star-shaped transition systems is a solution to this problem.