A new criterion for $$\mathcal {M}, \mathcal {N}$$-adhesivity, with an application to hierarchical graphs

Abstract

Adhesive categories provide an abstract framework for the algebraic approach to rewriting theory, where many general results can be recast and uniformly proved. However, checking that a model satisfies the adhesivity properties is sometimes far from immediate. In this paper we present a new criterion giving a sufficient condition for $$\mathcal {M},\mathcal {N}$$ M , N -adhesivity, a generalisation of the original notion of adhesivity. We apply it to several existing categories, and in particular to hierarchical graphs, a formalism that is notoriously difficult to fit in the mould of algebraic approaches to rewriting and for which various alternative definitions float around.

Fundamentals of Compositional Rewriting Theory

“…As already presented in Section 4.1 and 4.2, at present there already exists a certain amount of mathematical methodology in this regard, most notably comma category constructions of categories with adhesivity properties, and a variant thereof, so-called Artin gluing [7], for constructing quasi-topoi (cf. also [44] for recent advances in constructing (M,N)-adhesive categories). On the other hand, as summarized in Table 3, we are left to wonder whether it is indeed adhesivity properties that most generally characterize categories suitable for SqPO-semantics, since this semantics does not require the full-fledged variant of the relevant van Kampen square axioms (i.e., only axioms (X-iii-a), but not axioms (X-iii-b)), while on the other hand in order to support compositional SqPO semantics, it is required that the underlying category has pullbacks, and that it has FPCs along M-morphisms.…”

“…Overall, and as discussed throughout the paper, but in particular in full detail in Section 4, our approach to determining suitable classes of categories supporting the various rewriting semantics relies heavily on the categorical rewriting and category theory literature, such as the traditional framework of Ehrig et al that is based upon the notion of M-adhesive categories [3], but also a large number of works on a variety of other categories with adhesivity properties [5,4,40,38,41,6,35,36,20,7]. As discussed in further detail in Example 2 and in Section 5, there exist practically relevant application examples of rewriting theories that require to generalize the categorical formalism even further, e.g., to (M,N)-adhesive categories [42,43], of which in particular the category SGraph of directed simple graphs was recently found [44] to provide a non-trivial example (cf. the discussion in Example 2 for further details).…”

“…Many other examples concern comma category constructions, with a number of illustrative examples provided in Table 1. More intricate examples still have been developed in the context of so-called hierarchical graphs, which are obtained via commacategorical constructions along various notions of super-power functors, and whose adhesivity properties have been studied in [56,57] (see also [44]). 2) undirected multigraphs [29] *…”

“…It is worthwhile emphasizing however that there might exists another possible route towards a resolution of the problem regarding failure of compositional generic DPO-semantics for the category SGraph and a number of other practically relevant examples: according to recent results of [44,Thm. 3.1], SGraph (referred to as "DGraph" in loc.…”

“…Moreover, according to Lemma 10, the assumptions on C suffice to demonstrate that C has FPAs, hence the notions of admissible matches of rules and of composite rules are also well-posed. The rest of the proof then follows by instantiating (44) for the case of generic SqPO-semantics. In particular, the explicit formula for the rule composition is obtained by taking advantage of the results of Theorem 15, i.e., noting that the source functor S : D 1 → D 0 of the compositional rewriting double category is a multi-opfibration, and that the target functor T : D 1 → D 0 is a residual multi-opfibration.…”