Higher-dimensional categories with finite derivation type

Yves Guiraud and Philippe Malbos

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.


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