Section: New Results

Homotopy of rewriting systems

Participants : Cyrille Chenavier, Pierre-Louis Curien, Yves Guiraud, Maxime Lucas, Philippe Malbos, Jovana Obradović.

Coherent presentations of Artin monoids

With Stéphane Gaussent (ICJ, Univ. de Saint-Étienne), Yves Guiraud and Philippe Malbos have used higher-dimensional rewriting methods for the study of Artin monoids, a class of monoids 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 monoid  [65] , and one by Deligne about the actions of Artin monoids on categories  [47] , both proved by geometrical 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 and to extend both theorems. This work will appear in Compositio Mathematica [18] and has been implemented in a Python library.

The next objective of this collaboration is to extend those results in every dimension, first to Artin monoids, then to Artin groups, with a view towards two well-known open problems in the field: the word problem of Artin groups and the so-called $K(\pi ,1)$ conjecture.

New methods for the computation of polygraphic resolutions

Maxime Lucas, supervised by Pierre-Louis Curien and Yves Guiraud, develops Squier's theory in the setting of cubical $\omega$-categories. This will allow easier and more explicit computations of polygraphic resolutions than in the globular setting of [5] , and the use of new effective methods such as the reversing algorithm from Garside theory  [46] .

Yves Guiraud currently collaborates with Patrick Dehornoy (Univ. de Caen) and Matthieu Picantin (LIAFA, Univ. Paris 7) to extend the constructions of [18] to other important families of monoids, such as the plactic monoid, the Chinese monoid and the dual braid monoids.

Higher-dimensional linear rewriting

Cyrille Chenavier, Pierre-Louis Curien, Yves Guiraud and Philippe Malbos investigate with Eric Hoffbeck (LAGA, Univ. Paris 13) and Samuel Mimram (LIX, École Polytechnique) the links between set-theoretic rewriting theory and the computational methods known in symbolic algebra, such as Gröbner bases  [39] . This interaction is supported by the Focal project of the IDEX Sorbonne Paris Cité.

With Eric Hoffbeck (LAGA, Univ. Paris 13), Yves Guiraud and Philippe Malbos have introduced the setting of linear polygraphs to formalise a theory of linear rewriting (in the sense of linear algebra), generalising Gröbner bases. They have adapted to algebras the procedure of [5] that computes polygraphic resolutions from convergent presentations of monoids, with applications to the decision of an important homological property called Koszulness. This work is contained in [35] and it has been presented at IWC 2014 [31] .

Cyrille Chenavier, supervised by Yves Guiraud and Philippe Malbos, explores the use of Berger's theory of reduction operators  [38] to design new methods for the study of linear rewriting systems, and to promote the use of rewriting techniques in combinatorial algebra.

Homotopical and homological finiteness conditions

Yves Guiraud and Philippe Malbos have written a comprehensive introduction [36] on the links between higher-dimensional rewriting, the homotopical finiteness condition “finite derivation type” and the homological finiteness condition “${\mathrm{FP}}_3$”, from the point of view of higher categories and polygraphs. The purpose of this work is to provide an introduction to the field, formulated in a contemporary language, and with new, more formal proofs of classical results.

Wiring structure of operads and operad-like structures

Building on recent ideas of Marcelo Fiore on the one hand, and of François Lamarche on the other hand, Pierre-Louis Curien and Jovana Obradović develop a syntactic approach, using some of the kit of Curien-Herbelin's duality of computation and its polarised versions of Munch and Curien, to the definition of various structures that have appeared in algebra under the names of operads, cyclic operads, dioperads, properads, modular and wheeled operads, permutads, etc.... These structures are defined in the literature in different flavours. We seek to formalise the proofs of equivalence between these different styles of definition, and to make these proofs modular, so as not to repeat them for each variation of the notion of operad. Preliminary results are being presented in January 2015 at the Mathematical Institute of the Academy of Sciences (Belgrade).