String Diagrams for Non-Strict Monoidal Categories

Authors Paul Wilson , Dan Ghica, Fabio Zanasi

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Author Details

Paul Wilson
  • University of Southampton, UK
Dan Ghica
  • University of Birmingham, UK
Fabio Zanasi
  • University College London, UK

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Paul Wilson, Dan Ghica, and Fabio Zanasi. String Diagrams for Non-Strict Monoidal Categories. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 37:1-37:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Whereas string diagrams for strict monoidal categories are well understood, and have found application in several fields of Computer Science, graphical formalisms for non-strict monoidal categories are far less studied. In this paper, we provide a presentation by generators and relations of string diagrams for non-strict monoidal categories, and show how this construction can handle applications in domains such as digital circuits and programming languages. We prove the correctness of our construction, which yields a novel proof of Mac Lane’s strictness theorem. This in turn leads to an elementary graphical proof of Mac Lane’s coherence theorem, and in particular allows for the inductive construction of the canonical isomorphisms in a monoidal category.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • String Diagrams
  • Strictness
  • Coherence


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