Categorical Abstract Rewriting Systems and Functoriality of Graph Transformation

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.