Open Diagrams via Coend Calculus (Román, 2020)

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Abstract. Morphisms in a monoidal category are usually interpreted as processes, and graphically depicted as square boxes. In practice, we are faced with the problem of interpreting what non-square boxes ought to represent in terms of the monoidal category and, more importantly, how should they be composed. Examples of this situation include lenses or learners. We propose a description of these non-square boxes, which we call open diagrams, using the monoidal bicategory of profunctors. A graphical coend calculus can then be used to reason about open diagrams and their compositions.

Coend calculus has a graphical syntax on the monoidal bicategory of profunctors. The point of coend calculus is to compute a particular element; thus, we upgrade it to the monoidal bicategory of pointed profunctors. This explains why open diagrams appear and are used across the literature and why open internal diagrams have a good semantics in terms of profunctors.

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@inproceedings{roman21:openDiagrams,
  author    = {Mario Rom{\'{a}}n},
  editor    = {David I. Spivak and
               Jamie Vicary},
  title     = {Open Diagrams via Coend Calculus},
  booktitle = {Proceedings of the 3rd Annual International Applied Category Theory
               Conference 2020, {ACT} 2020, Cambridge, USA, 6-10th July 2020},
  series    = {{Electronic Proceedings in Theoretical Computer Science}},
  publisher = {Open Publishing Association},
  volume    = {333},
  pages     = {65--78},
  year      = {2021},
  month     = {Feb},
  doi       = {10.4204/EPTCS.333.5},
  ISSN      = {2075-2180},
}