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 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.
# How to cite