Coend Calculus and Open Diagrams
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Morphisms in a monoidal category are usually interpreted as processes or black boxes that can be composed sequentially and in parallel. In practice, we are faced with the problem of interpreting what non-square boxes ought to represent 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, in terms of coends and the monoidal bicategory of profunctors, with features of what could be considered a graphical calculus for these coends. The graphical calculus allows us to describe possible compositions of these open diagrams but also to reason about their concrete descriptions. This is work in progress.
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