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Constructing monoidal theories with open-graphs and rewrite categories

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In this talk, I’ll define open-graphs, which are special kinds of typed directed graphs that are well suiting for constructing free monoidal categories. These are best thought of as discretisations of polarised topological graphs, which Joyal and Street used to formalise graphical languages for monoidal categories in 1991. As in topological graphs, edges in open-graphs can be disconnected at one or both ends (forming inputs and outputs) and can be connected to themselves (forming circles). However, unlike topological graphs, open-graphs are discete, finitary, and well-suited to computational applications using existing techniques in graph rewriting. I’ll discuss how rewriting can be performed using the “double pushout” technique in the ambient adhesive category of typed graphs and show how open-graphs modulo certain rewrite systems can be used to construct free monoidal categories, PRO Ps, and more general monoidal theories. If there is time, I’ll discuss how we are applying these techniques to the study of many-body quantum entanglement.

This talk is part of the Category Theory Seminar series.

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