# Open Diagrams via Coend Calculus

arXiv: https://arxiv.org/abs/2004.04526 · HTML: https://mlsanity.com/paper/2004.04526/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2020.

## Built on by (most cited first, 3 of 3)

- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Towards Compositional Interpretability for XAI](https://mlsanity.com/paper/2406.17583/) (2024)
- [On the Anatomy of Attention](https://mlsanity.com/paper/2407.02423/) (2024)

## Abstract (arXiv)

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.