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, generalising the one introduced by Squier for word rewriting systems. We characterise 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.
Keywords: n-category, rewriting, polygraph, finite derivation type, low-dimensional topology
2000 MSC: 18C10, 18D05, 18D10, 57M20, 68Q42
Theory and Applications of Categories,
Vol. 22, 2009,
No. 18, pp 420-478.