A simple criterion for M,N-adhesivity

Adhesive categories, and variants such as M,N-adhesive ones, marked a watershed moment for the algebraic approaches to the rewriting of graph-like structures, since they provide an abstract framework where many general results (on e.g. parallelism) could be recast and uniformly proved. However, often checking that a model satisfies the adhesivity properties is far from immediate. In this paper we present a new criterion giving a sufficient condition for M,N-adhesivity, a generalisation of the original notion of adhesivity. To show the effectiveness of this criterion, we apply it to several existing categories of graph-like structures, including hypergraphs, various kinds of hierarchical graphs (a formalism that is notoriously difficult to fit in the mould of algebraic approaches to rewriting), and combinations of them.