String diagrams for Strictification and Coherence

TL;DR

This paper develops a presentation of string diagrams for non-strict monoidal categories, providing new graphical tools that facilitate applications in computer science and offering elementary proofs of fundamental theorems.

Contribution

It introduces a generators-and-relations presentation for non-strict monoidal string diagrams and proves their correctness, leading to elementary proofs of Mac Lane's strictness and coherence theorems.

Findings

Provides a graphical formalism for non-strict monoidal categories.

Offers a new proof of Mac Lane's strictness theorem.

Enables inductive construction of canonical isomorphisms.

Abstract

Whereas string diagrams for strict monoidal categories are well understood, and have found application in several fields of Computer Science, graphical formalisms for non-strict monoidal categories are far less studied. In this paper, we provide a presentation by generators and relations of string diagrams for non-strict monoidal categories, and show how this construction can handle applications in domains such as digital circuits and programming languages. We prove the correctness of our construction, which yields a novel proof of Mac Lane's strictness theorem. This in turn leads to an elementary graphical proof of Mac Lane's coherence theorem, and in particular allows for the inductive construction of the canonical isomorphisms in a monoidal category.