Combinatorics of past-similarity in higher dimensional transition systems

TL;DR

This paper explores the concept of past-similarity in higher-dimensional transition systems, linking it to model category structures, fibrancy, and weak equivalences, and discusses its implications for causality and homotopy.

Contribution

It introduces the notion of past-similarity in higher-dimensional transition systems and characterizes fibrant objects and weak equivalences in this context.

Findings

Past-similarity is key to understanding model categories of transition systems.

Fibrant objects have transition sets closed under past-similarity.

Weak equivalences are characterized by isomorphisms after identifying past-similar states.

Abstract

The key notion to understand the left determined Olschok model category of star-shaped Cattani-Sassone transition systems is past-similarity. Two states are past-similar if they have homotopic pasts. An object is fibrant if and only if the set of transitions is closed under past-similarity. A map is a weak equivalence if and only if it induces an isomorphism after the identification of all past-similar states. The last part of this paper is a discussion about the link between causality and homotopy.