The geometry of cubical and regular transition systems

TL;DR

This paper explores the structure of cubical and regular transition systems, providing categorical and homotopical insights, and offers a combinatorial description of fibrant systems along with a general lemma on adjunctions.

Contribution

It introduces a detailed combinatorial characterization of fibrant cubical and regular transition systems and discusses their categorical and homotopical properties.

Findings

Existence of cubical transition systems with arbitrarily large cubes.

Regular transition systems have similar categorical and homotopical properties to cubical ones.

A complete combinatorial description of fibrant systems is provided.

Abstract

There exist cubical transition systems containing cubes having an arbitrarily large number of faces. A regular transition system is a cubical transition system such that each cube has the good number of faces. The categorical and homotopical results of regular transition systems are very similar to the ones of cubical ones. A complete combinatorial description of fibrant cubical and regular transition systems is given. One of the two appendices contains a general lemma of independant interest about the restriction of an adjunction to a full reflective subcategory.