The task of constructing compositional semantics for network-style diagrammatic languages, such as electrical circuits or chemical reaction networks, has been dubbed the black boxing problem, as it gives semantics that describes the properties of each network that can be observed externally, through composition, while discarding the internal structure. One way to solve these problems is to formalise the diagrams and their semantics using hypergraph categories, with semantic interpretation a hypergraph functor, called the black box functor, between them. Building on a previous method for constructing hypergraph categories and functors, known as decorated corelations, in this paper we construct a category of decorating data, and show that the decorated corelations method is itself functorial, with a universal property characterised by a left Kan extension. We then show that any hypergraph category can be presented in terms of decorating data, and hence argue that the category of decorating data is a good setting in which to construct any hypergraph functor. As an example, we give a new construction of Baez and Pollard's black box functor for reaction networks.
Keywords: decorated corelation, Frobenius monoid, hypergraph category, black box functor, well-supported compact closed category
2020 MSC: 18M35, 18M30, 18B10
Theory and Applications of Categories, Vol. 35, 2020, No. 26, pp 979-1011.