Section: New Results

Effective higher dimensional algebra

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

Rewriting methods for Artin monoids

With Stéphane Gaussent (ICJ, Univ. 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 formulates 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  [76] , and one by Deligne about the actions of Artin monoids on categories  [58] , both originally proved by geometrical and non-constructive methods. An improved completion procedure, called the homotopical completion-reduction procedure (see also [8] ), is formalised and used to give constructive proofs of (improved versions of) both theorems. This work has been published in Compositio Mathematica [19] and has been implemented in a Python library (http://www.pps.univ-paris-diderot.fr/~guiraud/cox/cox.zip ).

Rewriting and Garside theory

Yves Guiraud has collaborated with Patrick Dehornoy (LNO, Univ. Caen) to develop an axiomatic setting for monoids with a special notion of quadratic normalisation map with good computational properties. This theory generalises the normalisation procedure known for monoids that admit a special family of generators called a Garside family  [57] to a much wider class that also includes the plactic monoids. It is proved that good quadratic normalisation maps correspond to quadratic convergent presentations, together with a sufficient condition for this to happen, based on the shape of the normalisation paths on length-three words. This work has been submitted for publication to the Journal de l'École Polytechnique — Mathématiques [44] .

Building on this last article, Yves Guiraud currently collaborates with Matthieu Picantin (Automates team, LIAFA, Univ. Paris 7) to generalise the main results of [19] to monoids with a Garside family. This will allow an extension of the field of application of the rewriting methods to other geometrically interesting classes of monoids, such as the dual braid monoids.

Higher-dimensional linear rewriting

With Eric Hoffbeck (LAGA, Univ. Paris 13), Yves Guiraud and Philippe Malbos have introduced in  [64] the setting of linear polygraphs to formalise a theory of linear rewriting, generalising Gröbner bases. They have adapted the computational method of [7] to compute polygraphic resolutions of associative algebras, with applications to the decision of the Koszul homological property. They are currently engaged into a major overhaul of this work, whose main goal is to ease the adaptation of the results to other algebraic varieties, like commutative algebras or Lie algebras.

Theory of reduction operators

Cyrille Chenavier, supervised by Yves Guiraud and Philippe Malbos, explores the use of Berger's theory of reduction operators  [50] to design new rewriting methods in algebra. In [42] , he proposed a construction of a contracting homotopy for the Koszul complex of an algebra (a complex characterising the homological property of Koszulness): when an algebra admits a side-confluent presentation (a strong hypothesis of confluence), he gave a candidate for the contracting homotopy, built using specific representations of confluence algebras; when the presentation satisfies an additional condition, called the extra-condition, it turns out that this candidate works.

Rewriting methods for coherence

In [45] , Maxime Lucas, supervised by Yves Guiraud and Pierre-Louis Curien, has applied the rewriting techniques of  [65] to prove coherence theorems for bicategories and pseudofunctors. He obtained a coherence theorem for pseudonatural transformations thanks to a new theoretical result, improving on the former techniques, that relates the properties of rewriting in 1- and 2-categories.

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 developed a syntactic approach, using some of the kit of Curien-Herbelin’s duality of computation and its polarised versions by 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. The goal is to formalise the proofs of equivalence between these different styles of definition. This work is completed for cyclic operads and was presented at the conference Category Theory 2015 in Aveiro [43] . Further work will be to make these proofs modular, so as not to repeat them for each variation of the notion of operad.

A graphical proof of the Bénabou-Roubaud theorem

As a substantial development of reasoning with string diagrams, Jovana Obradović gave a complete proof of the Bénabou-Roubaud monadic descent theorem in [47] . One of the essential points concerning Grothendieck's original approach to descent theory consists of identifying the class of effective descent morphisms for a given fibration. In the special case of a bifibration satisfying Beck-Chevalley condition, Bénabou and Roubaud have given such a characterisation by means of monadicity. Due to the technically complicated calculations involving Grothendieck's cocycle condition, the categorical equivalence which reflects the comparison of the descent in fibered categories with monadic descent is usually not worked out in complete detail in the literature. Jovana Obradović linked the monadic and the original viewpoint via another possible definition of the category of descent data. This intermediate step, due to Janelidze and Tholen, involves constructions in internal categories and it provides an example on how one can stay in the world of string diagrams even when dealing with morphisms which do not have an explicit string diagram definition.