Higher-dimensional categories with finite derivation type

TL;DR

This paper introduces the concept of finite derivation type for n-categories using polygraphs, generalizing Squier's property, and provides criteria and techniques for analyzing convergent presentations.

Contribution

It generalizes the finite derivation type concept to higher categories and offers new methods to study convergent presentations via polygraphs and critical branchings.

Findings

Defined finite derivation type for n-categories.

Characterized this property using critical branchings.

Provided techniques for analyzing convergent presentations.

Abstract

We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalizing the one introduced by Squier for word rewriting systems. We characterize this property by using the notion of critical branching. In particular, we define sufficient conditions for an n-category to have finite derivation type. Through examples, we present several techniques based on derivations of 2-categories to study convergent presentations by 3-polygraphs.