A Fibred Approach to Rewriting | How the Duality between Adding and DeletingCooperates with the Di erence between Matching

We present a new approach to rewriting obtained by enhancing and unifying existing variants inside the algebraic (or better categorical) approach to (graph) rewriting. Our approach is motivated by second-order term graph rewriting and stresses on one hand the two-step nature of rule application consisting of deleting and adding items and on the other hand the heterogeneous nature of the rewriting setup where rule steps should be clearly distinguished from matching of rule sides into redexes. Complementing the existing oppbration approach with a dual bration step turns out to yield a natural and exible approach with useful new applications. The resulting bred approach takes advantage of the heterogeneous setting and appropriately reeects the duality between deleting and adding in the course of rewriting, in contrast with the double-pushout approach which simpliies this duality into a symmetry. An important contribution is the universal characterisation of the host object, which has to be found as a pushout-complement in the double pushout approach. The bred approach is presented in abstract and independent from any concrete application categories in the manner of High-Level Replacement Systems. Our original motivation for the development of the bred approach comes from term graphs with bound variables where all other approaches failed; in this paper we present an unusual view on term rewriting as running example.