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.
We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are wellbehaved. Many types of graphical structures used in computer science are shown to be examples of adhesive categories. Double-pushout graph rewriting generalises well to rewriting on arbitrary adhesive categories.
We introduce the theory IH R of interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IH R are derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid-comonoid pairs. This construction is instrumental in showing that IH R is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R.
We introduce IH, a sound and complete graphical theory of vector subspaces over the field of polynomial fractions, with relational composition. The theory is constructed in modular fashion, using Lack's approach to composing PROPs with distributive laws. We then view string diagrams of IH as generalised stream circuits by using a formal Laurent series semantics. We characterize the subtheory where circuits adhere to the classical notion of signal flow graphs, and illustrate the use of the graphical calculus on several examples.
International audienceNetwork theory uses the string diagrammatic language of monoidalcategories to study graphical structures formally, eschewing specialisedtranslations into intermediate formalisms. Recently, therehas been a concerted research focus on developing a network theoreticapproach to signal flow graphs, which are classical structuresin control theory, signal processing and a cornerstone in the studyof feedback. In this approach, signal flow graphs are given a relationaldenotational semantics in terms of formal power series.Thus far, the operational behaviour of such signal flow graphshas only been discussed at an intuitive level. In this paper we equipthem with a structural operational semantics. As is typically thecase, the purely operational picture is too concrete – two graphsthat are denotationally equal may exhibit different operational behaviour.We classify the ways in which this can occur and showthat any graph can be realised – rewritten, using the graphical theory,into an executable form where the operational behavior and thedenotation coincides
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