Adhesive and quasiadhesive categories

Abstract

Abstract. We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved, as well as quasiadhesive categories which restrict attention to regular monomorphisms. Many examples of graphical structures used in computer science are shown to be examples of adhesive and quasiadhesive categories. Double-pushout graph rewriting generalizes well to rewriting on arbitrary adhesive and quasiadhesive categories.

Confluence of Graph Rewriting with Interfaces

“…Our key observation is that for DPO rewriting with interfaces, the Knuth-Bendix property holds and therefore confluence of a terminating system can be decided by checking whether its critical pairs are joinable. More precisely, if some mild assumptions related to the computability of performing rewriting steps on the underlying notion of term are satisfied, our result holds for the most general venue available for DPO rewriting, namely, adhesive categories [34].…”

“…The arrows of Hyp are homomorphisms: functions G → H such that for each k, l, G k,l → H k,l they respect the source and target maps in the obvious way. The seasoned reader will recognise Hyp as a presheaf topos, and as such, it is adhesive [34]. We shall visualise hypergraphs as follows: is a node and is an hyperedge, with ordered tentacles attached to the left boundary linking to sources and those on the right linking to targets.…”

“…In order not to restrict ourselves to any one concrete model of graphs, we work with adhesive categories [34]. Adhesive categories are relevant because they have well-behaved pushouts along monomorphisms, and for this reason they are convenient as ambient categories for DPO rewriting.…”

Collagories for Relational Adhesive Rewriting

“…Further investigations will explore different variations of adhesive categories in a collagory setting, including the quasiadhesive categories of [LS05], and their applications.…”

“…Adhesive categories as a more specific setting for double-pushout graph rewriting have been introduced by Lack and Sobociński [LS04,LS05]; the following two definitions are taken from there:…”