A recipe for black box functors

TL;DR

This paper develops a categorical framework using decorated corelations to construct hypergraph functors, providing a systematic method for creating black box semantics for network diagrams like electrical circuits and reaction networks.

Contribution

It introduces a category of decorating data and proves the functoriality of decorated corelations, enabling systematic construction of hypergraph functors including black box semantics.

Findings

Constructed a category of decorating data for hypergraph functors.

Proved decorated corelations form a functor with a universal property.

Reconstructed Baez and Pollard's black box functor for reaction networks.

Abstract

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, by 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. Reviewing a principled method for constructing hypergraph categories and functors, known as decorated corelations, in this paper we construct a category of \emph{decorating data}, and show that the decorated corelations method is itself functorial, with a universal property characterised by a left Kan extension. We then argue that the category of decorating data is a good setting in which to construct any hypergraph functor, giving a new construction of Baez and Pollard's black box functor for reaction networks as an example.