Representing 2-Dimensional Critical Pairs

Polygraphs generalize to n-categories the usual notion of equational theory, thus allowing one to describe a category by the means of generators and relations. When the relations are oriented, such a presentation can be considered as a rewriting system and one might wonder whether the rewriting system is confluent and terminating in order to provide a notion of canonical representative of morphisms modulo equations (the normal forms of the morphisms). In term rewriting systems, confluence is often proved by computing the critical pairs, which are in finite number, and showing that they are joinable. We extend here this methodology to polygraphs presenting 2-categories. This task is not straightforward because a finite polygraph might admit an infinite number of critical pairs. This leads us to introduce the multicategory of contexts of the free compact 2-category generated by a 2-category, in which we can embed the original 2-category generated by the polygraph and compute a finite number of morphisms which generate all the critical pairs. We also introduce polygraphic nets, which are a concrete representation of contexts. These theoretical tools allow us to finally describe an algorithm for computing generating families of critical pairs in 2-dimensional polygraphs. Term rewriting systems have proven very useful to reason about terms modulo equations. In some cases, the equations can be oriented and completed in a way giving rise to a convergent (that is both confluent and terminating) rewriting system, thus providing a notion of canonical representative of equivalence classes of terms. Usually, the terms are freely generated by a signature (Σn)n∈N, which consists of a family of sets Σn of generators of arity n, and one considers equational theories on such a signature, which consist of equations formalized by pairs of terms freely generated by the signature. For example, the equational theory of monoids contains two generators m and e, whose arities are respectively 2 and 0, and three equations m(m(x, y), z) = m(x,m(y, z)) m(e, x) = x and m(x, e) = x (1) ∗CEA LIST, Laboratory for the Modelling and Analysis of Interacting Systems, Point Courrier 94, 91191 Gif-sur-Yvette, France. E-mail: [email protected]. This work was started while I was in the PPS team (CNRS – Univ. Paris 7) and has been supported by the CHOCO (“Curry Howard pour la Concurrence”, ANR-07-BLAN-0324) French ANR.