A survey of graphical languages for monoidal categories

TL;DR

This survey provides a comprehensive overview of graphical languages for monoidal categories, aiming to serve as a reference for mathematicians, physicists, and computer scientists interested in diagrammatic reasoning.

Contribution

It compiles and explains various notions of monoidal categories and their string diagrams, offering an accessible, informal guide with references for further study.

Findings

Summarizes different graphical languages for monoidal categories

Highlights connections between diagrammatic reasoning and mathematical structures

Provides pointers to rigorous treatments in literature

Abstract

This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs. Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study. Where possible, we provide pointers to more rigorous treatments in the literature. Where we include results that have only been proved in special cases, we indicate this in the form of caveats.