Open Diagrams via Coend Calculus (Román, 2020)
 Article: pdf.
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 nonsquare 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 nonsquare 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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