Categorical rewriting of term-like structures

The article surveys a recent series of papers by the authors investigating the categorical foundations of various rule-based formalisms. The starting point is the well-known representation of term rewriting systems as cartesian 2-categories, based on the characterization of finite terms as arrows of a Lawvere theory. We first show that many term-like structures (including cyclic term graphs, mu-terms and rational terms) can be characterized as arrows of suitable theories. Next we represent rules as cells over a theory, and we show that the free 2-category generated by these cells faithfully represents the rewrite sequences of the original rewriting system.