Open Diagrams via Coend Calculus

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.

Formatting revision after Proceedings ACT 2020, minor changes

References in corpus (6)

• Quantum Circuits Architecture
• Quasistrict symmetric monoidal 2-categories via wire diagrams
• Cartesian Bicategories II
• Constructing symmetric monoidal bicategories functorially
• Profunctor Optics: Modular Data Accessors
• Comb Diagrams for Discrete-Time Feedback

Cited by in corpus (15)

• Profunctor Optics, a Categorical Update
• Bayesian Updates Compose Optically
• Category Theory in Machine Learning
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• Quantum Supermaps are Characterized by Locality
• Coend Optics for Quantum Combs
• Cyber Kittens, or Some First Steps Towards Categorical Cybernetics
• String Diagrams for Layered Explanations
• Cornering Optics
• Causality in Higher Order Process Theories
• Collages of String Diagrams
• Optics for Premonoidal Categories
• A dataflow programming framework for linear optical distributed quantum computing
• Combs, Causality and Contractions in Atomic Markov Categories
• A Profunctorial Semantics for Quantum Supermaps