Towards 3-Dimensional Rewriting Theory

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 confluent 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 confluence for polygraphs presenting 2-categories and introduce a framework in which a finite 3-dimensional rewriting system admits a finite number of critical pairs. Recent developments in category theory have established higher-dimensional categories as a fundamental theoretical setting in order to study situations arising in various areas of mathematics, physics and computer science. A nice survey of these can be found in [2], explaining how the use of category theory enables one to unify these apparently unrelated fields of science, by revealing that their intrinsic algebraic structures are in fact closely connected. In the last decade, higher-dimensional categories have therefore emerged as a tool of everyday use for many scientists. The motivation behind the concept of higher dimensions here is that, in order to have a fine-grained understanding of the algebraic structures at stake, one should not only consider morphisms between objects involved, but also morphisms between morphisms (i.e. 2-dimensional morphisms), morphisms between morphisms between morphisms (i.e. 3-dimensional morphisms), and so on. For example, the starting point of algebraic topology [16] is that one should not consider points and paths between them in topological spaces, but also homotopies between paths, and can be refined by also considering homotopies between homotopies and so on. The categorical structures considered nowadays are thus becoming more and more complex, which enables them to capture many details, but the proofs are becoming more and more complicated too, and we are facing the urge for new tools, both of a theoretical and practical nature, in order to make them easier and more manageable. In particular, many proofs of even conceptually simple facts involve showing the commutativity of diagrams This work was partially funded by the french ANR project CATHRE ANR-13-BS02-0005-02. This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.