Section: New Results

Homotopy of rewriting systems

Participants : Pierre-Louis Curien, Yves Guiraud, Philippe Malbos.

Coherence in monoidal structures

Yves Guiraud and Philippe Malbos have applied the Squier's homotopical theorem  [70] , which they had generalised to higher-dimensional rewriting systems  [52] , to several types of categories with monoidal structures. This work develops a formal setting to produce constructive proofs of coherence conditions, applied to the cases of monoidal categories, symmetric monoidal categories and braided categories. These results have been published in Mathematical Structures in Computer Science [12] .

Computation of resolutions of monoids

Yves Guiraud and Philippe Malbos have extended Squier's homotopical theory to the higher dimensions of presentations of monoids to get an algorithm transforming a convergent word rewriting system into a polygraphic resolution of the presented monoid, in the setting of Métayer  [63] . From this polygraphic resolution, this work gives an explicit procedure to recover several of the known Abelian resolutions of the monoid, generalising and relating algebraic invariants of monoids. Moreover, a polygraphic resolution corresponds to the normalisation strategies of rewriting systems and they contain all the proofs of equality between elements, together with the proofs of equality of those proofs of equality, and so on. This work has been published in Advances in Mathematics [13] . By nature, polygraphic resolutions bear many similarities with the higher-dimensional groupoids that appear in homotopical type theory when one considers the towers of identity types: this connection will be investigated by Pierre-Louis Curien, Yves Guiraud, Hugo Herbelin and Matthieu Sozeau.

Coherent presentations of Artin groups

With Stéphane Gaussent (Institut Camille Jordan, Université de Saint-Étienne), Yves Guiraud and Philippe Malbos are currently finishing an article on the rewriting properties and coherence issues in Artin groups, a class of groups that is fundamental in algebra and geometry. This work uses the formal setting of coherent presentations (a truncation of polygraphic resolutions at the level above relations) to formulate, in a common language, several known results in combinatorial group theory: one by Tits about the fundamental group of a graph associated to an Artin group  [71] , and one by Deligne about the actions of Artin groups on categories  [44] , both proved by geometrical and non-constructive methods. In this work, an improvement of Knuth-Bendix's completion procedure is introduced, called the homotopical completion-reduction procedure, and it is used to give a constructive proof of both those theorems. In fact, the method even improves the formerly known results: for example, it generalises Deligne's result to cases where his geometrical proof cannot be applied. A preliminary version of this work is available online [31] . The next objective of this collaboration is to extend the formal setting and methods to compute polygraphic resolutions of Artin groups, with a view towards two open problems of combinatorial group theory with respect to Artin groups: the decidability of the word problem and the verification that a precise topological space is a classifying space.

Higher-dimensional linear rewriting

With Samuel Mimram (CEA Saclay) and Pierre-Louis Curien, Yves Guiraud and Philippe Malbos investigate the links between set-theoretic rewriting theory and the computational methods known in symbolic algebra, such as Gröbner bases  [36] . In particular, this work is interested in extending the setting of higher-dimensional rewriting to include “linear rewriting” and, as a consequence, to be able to apply its methods in symbolic computation. One particular direction is to understand Anick's resolution  [33] , and to improve it with the completion-reduction methodology, in order to get better algorithms to compute homological invariants and to prove important properties such as Koszulness. This research direction has been presented to the first call for projects of the IDEX Sorbonne-Paris-Cité, together with Eric Hoffbeck and Muriel Livernet (LAGA, Université Paris 13) and François Métayer (PPS, Université Paris 7).