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Nominal String Diagrams

Authors Samuel Balco, Alexander Kurz

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Author Details

Samuel Balco
  • Department of Informatics,University of Leicester, United Kingdom
Alexander Kurz
  • Department of Computer Science, Chapman University, Orange California, USA


We are greatful to Fredrik Dahlqvist, Giuseppe Greco, Samuel Mimram, Drew Moshier, Alessandra Palmigiano, David Pym, Mike Shulman, Pawel Sobocinski, Thomas Streicher, Georg Struth, Apostolos Tzimoulis and Fabio Zanasi for discussions on the topic of this paper.

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Samuel Balco and Alexander Kurz. Nominal String Diagrams. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This requires us to take nominal sets as a monoidal category, not with the cartesian product, but with the separated product. To this end, we develop the beginnings of a theory of monoidal categories internal in a symmetric monoidal category. As an instance, we obtain a notion of a nominal PROP as a PROP internal in nominal sets. A 2-dimensional calculus of simultaneous substitutions is an application.

Subject Classification

ACM Subject Classification
  • Software and its engineering → General programming languages
  • Theory of computation → Models of computation
  • Theory of computation → Logic
  • Mathematics of computing
  • string diagrams
  • nominal sets
  • separated product
  • simultaneous substitutions
  • internal category
  • monoidal category
  • internal monoidal categories
  • PROP


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