Towards 3-dimensional rewriting theory

Abstract

String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even better, when the presentation is conuent and terminating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. One of the main purposes of this article is to give a progressive introduction to the notion of higher-dimensional rewriting system provided by polygraphs, and describe its links with classical rewriting theory, string and term rewriting systems in particular. After introducing the general setting, we will be interested in proving local conuence for polygraphs presenting 2-categories and introduce a framework in which a finite 3-dimensional rewriting system admits a finite number of critical pairs.