Categorified Path Calculus
Quantum Physics
2023-07-07 v1 Category Theory
Abstract
Path calculus, or graphical linear algebra, is a string diagram calculus for the category of matrices over a base ring. It is the usual string diagram calculus for a symmetric monoidal category, where the monoidal product is the direct sum of matrices. We categorify this story to develop a surface diagram calculus for the bicategory of matrices over a base bimonoidal category. This yields a surface diagram calculus for any bimonoidal category by restricting to diagrams for 1x1 matrices. We show how additional structure on the base category, such as biproducts, duals and the dagger, adds structure to the resulting calculus. Applied to categorical quantum mechanics this yields a new graphical proof of the teleportation protocol.
Cite
@article{arxiv.2307.03075,
title = {Categorified Path Calculus},
author = {Simon Burton},
journal= {arXiv preprint arXiv:2307.03075},
year = {2023}
}