Categorical Abstract Rewriting Systems and Functoriality of Graph\n Transformation

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