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Hypergraph categories as cospan algebras

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If you have a question about this talk, please contact Jamie Vicary.

Hypergraph categories are symmetric monoidal categories in which morphisms can be represented by string diagrams in which strings can branch and split: diagrams that are reminiscent of electrical circuits. As such they provide a framework for formalising the syntax and semantics of circuit-type diagrammatic languages. This structure has been independently rediscovered many times, in contexts as diverse as concurrency theory, databases, signal flow graphs and linear algebra, graph rewriting, and circuit theory. Nonetheless, despite its utility, the definition has the perhaps intriguing property that hypergraph structure does not transfer along equivalence of categories.

In this talk I will motivate the definition by providing an alternative conceptualisation of a hypergraph category as a so-called cospan algebra. A cospan algebra is a lax symmetric monoidal functor (Cospan_L,+)—> (Set,x), where L is some fixed set, and Cospan_L is the category where objects are finite sets X equipped with a function X—> L, and morphisms are cospans between finite sets that respect these types. In particular, the category of cospan algebras is equivalent to the category of strict hypergraph categories.

I will also make some remarks on constructions of free hypergraph categories, and explain how this perspective relates to decorated cospans and corelations. Only a basic familiarity with monoidal categories will be assumed.

This is joint work with David Spivak and Maru Sarazola.

This talk is part of the Theoretical computer science seminar series.

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