Cornering Optics

TL;DR

This paper explores the mathematical structure of optics within monoidal categories, introducing the concept of free cornering to facilitate reasoning with comb diagrams and providing a graphical calculus for these constructs.

Contribution

It establishes that the category of optics naturally emerges from free cornering, offering a new framework and graphical tools for working with comb diagrams in monoidal categories.

Findings

Optics form a category within the free cornering of a monoidal category.

The free cornering provides an intuitive graphical calculus for optics and comb diagrams.

The framework enhances reasoning about complex diagrammatic structures in category theory.

Abstract

We show that the category of optics in a monoidal category arises naturally from the free cornering of that category. Further, we show that the free cornering of a monoidal category is a natural setting in which to work with comb diagrams over that category. The free cornering admits an intuitive graphical calculus, which in light of our work may be used to reason about optics and comb diagrams.