Here at the Simons Institute workshop on compositionality, my talk on network theory explained how to use ‘decorated cospans’ as a general model of open systems. These were invented by Brendan Fong, and are nicely explained in his thesis:
• Brendan Fong, The Algebra of Open and Interconnected Systems. (Blog article here.)
But he went further: to understand the externally observable behavior of an open system we often want to simplify a decorated cospan and get another sort of structure, which he calls a ‘decorated corelation’.
In this talk, Brendan explained decorated corelations and what they’re good for:
• Brendan Fong, Modelling interconnected systems with decorated corelations. (Talk slides here.)
Abstract. Hypergraph categories are monoidal categories in which every object is equipped with a special commutative Frobenius monoid. Morphisms in a hypergraph category can hence be represented by string diagrams in which strings can branch and split: diagrams that are reminiscent of electrical circuit diagrams. As such they provide a framework for formalising the syntax and semantics of circuit-type diagrammatic languages. In this talk I will introduce decorated corelations as a tool for building hypergraph categories and hypergraph functors, drawing examples from linear algebra and dynamical systems.
