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 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