Collages of string diagrams

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• Document: (arXiv).

Multiple extensions to string diagrams had been regarded as separate and ad-hoc, belonging to particular applications and using particular formalisms. This manuscript claims that this division is only apparent and that all these extensions are particular instances of the same encompassing idea: that of glueing multiple string diagrams into what we call a collage of string diagrams. We give semantics to collages in terms of bimodular profunctors.

Bimodular profunctors (or the version of Tambara modules apt to Bimodular categories) provide the necessary structure to consider independent pieces of string diagrams. This explains how the syntax for strong monoidal functor boxes works, but also the theory of open internal string diagrams. As a first example, we introduce string diagrams for bimodular categories: we use them for encoding a race condition in bimodular string diagrams.

Abstract. We introduce collages of string diagrams as a diagrammatic syntax for glueing multiple monoidal categories. Collages of string diagrams are interpreted as pointed bimodular profunctors. As the main examples of this technique, we introduce string diagrams for bimodular categories, string diagrams for functor boxes, and string diagrams for internal diagrams.

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@inproceedings{collages23,
  author       = {Dylan Braithwaite and
                  Mario Rom{\'{a}}n},
  editor       = {Sam Staton and
                  Christina Vasilakopoulou},
  title        = {Collages of String Diagrams},
  booktitle    = {Proceedings of the Sixth International Conference on Applied Category
                  Theory 2023, {ACT} 2023, University of Maryland, 31 July - 4 August
                  2023},
  series       = {{EPTCS}},
  volume       = {397},
  pages        = {39--53},
  year         = {2023},
  doi          = {10.4204/EPTCS.397.3},
}

See also

• Open Diagrams via Coend Calculus (Román, 2020) uses a first variant of collages to provide string diagrams for pointed profunctors. It justifies how these form “open diagrams”, or “string diagrams of string diagrams”.

Tags: Open diagram, bimodular tambara.