Categorical Abstract Rewriting Systems and Functoriality of Graph Transformation

TL;DR

This paper introduces a categorical framework for rewriting systems, defines functoriality (vertical composition), and demonstrates that most graph transformation systems are functorial, with some exceptions.

Contribution

It formalizes abstract rewriting in categories and establishes functoriality as a key property, providing insights into graph transformation systems.

Findings

Most graph transformation systems are functorial.

Counter-example of a non-functorial graph transformation system.

Introduces categorical notions to rewriting systems.

Abstract

Rewriting systems are often defined as binary relations over a given set of objects. This simple definition is used to describe various properties of rewriting such as termination, confluence, normal forms etc. In this paper, we introduce a new notion of abstract rewriting in the framework of categories. Then, we define the functoriality property of rewriting systems. This property is sometimes called vertical composition. We show that most of graph transformation systems are functorial and provide a counter-example of graph transformation systems which is not functorial.