Collagories: Relation-algebraic reasoning for gluing constructions

Abstract

The relation-algebraic approach to graph transformation has previously been formalised in the context of complete distributive allegories. Careful analysis reveals that the zero laws postulated for distributive allegories were never used, and that completeness was most importantly used for the difunctional closures necessary for a relation-algebraic characterisation of pushouts. We therefore define collagories essentially as “distributive allegories without zero morphisms”, and also define a variant of Kleene star to produce difunctional closures where necessary. Typical collagories relevant for generalised graph structure transformation can be obtained from basic collagories like that of sets and relations via nestable constructions of collagories of semi-unary algebras, which allow natural representations in particular of graph structures, also with fixed label sets, or with type graphs. Since collagories are intended as foundation for generalised graph structure transformation in the algebraic tradition, we concentrate particularly on co-tabulations, the core of the relation-algebraic gluing concept. We clarify the precise relationship between co-tabulations and pushouts, and investigate the special case of direct sums, which is particularly affected by the absence of zero laws. Finally, we consider Van Kampen squares, the central ingredient of the definition of adhesive categories that has recently become popular as foundation for algebraic graph transformation, and obtain an interesting characterisation of Van Kampen squares in collagories.