Hereditary Pushouts Reconsidered

Abstract

The introduction of adhesive categories revived interest in the study of properties of pushouts with respect to pullbacks, which started over thirty years ago in the category of graphs. Adhesive categories provide a single property of pushouts that suffices to derive lemmas that are essential for central theorems of double pushout rewriting such as the local Church-Rosser Theorem. The present paper shows that the same lemmas already hold for pushouts that are hereditary, i.e. those pushouts that remain pushouts when they are embedded into the associated category of partial maps. Hereditary pushouts - a twenty year old concept - induce a generalization of adhesive categories, which will be dubbed partial map adhesive. An application relevant category that does not fit the framework of adhesive categories and its variations in the literature will serve as an illustrating example of a partial map adhesive category.