Coherence in monoidal track categories

TL;DR

This paper develops homotopical rewriting methods in higher-dimensional categories to solve coherence problems in monoidal and braided monoidal categories, framing them as asphericity and word problems.

Contribution

It introduces a novel homotopical approach using rewriting on polygraphs to establish coherence results in complex algebraic categories.

Findings

Reformulates coherence as an asphericity problem in track categories.

Extends methods to braided monoidal categories.

Provides a systematic rewriting framework for higher-dimensional coherence.

Abstract

We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and we use rewriting methods on polygraphs to solve it. The setting is extended to more general coherence problems, seen as 3-dimensional word problems in a track category, including the case of braided monoidal categories.