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Section: New Results

Effective higher-dimensional algebra

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

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 [53] 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 published in the International Journal of Algebra and Computation [21].

Building on this last article, Yves Guiraud currently collaborates with Matthieu Picantin (IRIF, Univ. Paris 7) to generalise the main results of Gaussent, Guiraud and Malbos on coherent presentations of Artin monoids [7], 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.

Still in collaboration with Matthieu Picantin, Yves Guiraud develops an improvement of the classical Knuth-Bendix completion procedure, called the KGB completion procedure. The original algorithm tries to compute, from an arbitrary terminating rewriting system, a finite convergent presentation by adding relations to solve confluence issues. Unfortunately, this algorithm fails on standard examples, like most Artin monoids with their usual presentations. The KGB procedure uses the theory of Tietze transformations, together with Garside theory, to also add new generators to the presentation, trying to reach the convergent Garside presentation identified in [21]. The KGB completion procedure is partially implemented in the prototype Rewr, developed by Yves Guiraud and Samuel Mimram.

Higher-dimensional linear rewriting

With Eric Hoffbeck (LAGA, Univ. Paris 13), Yves Guiraud and Philippe Malbos have introduced in [65] the setting of linear polygraphs to formalise a theory of linear rewriting, generalising Gröbner bases. They have adapted the method of Guiraud and Malbos [9] to compute polygraphic resolutions of associative algebras, with applications to the decision of the Koszul homological property. They are currently finishing the major overhaul of this work, started in 2015, whose main goal is to ease the adaptation of the results to other algebraic varieties, like commutative algebras or Lie algebras.

Cyrille Chenavier, supervised by Yves Guiraud and Philippe Malbos, explored the use of Berger's theory of reduction operators [45] to improve the theory of Gröbner bases for associative algebras. This work has permitted to unveil two interesting algebraic structures that are hidden in rewriting theory. First, the operations that associate a normal form to an arbitrary word admit a structure of lattice, that gives a new algebraic characterisation of confluence and a new algorithm for completion, based on an iterated use of the meet-operation of the lattice. Second, under mild technical conditions, the different normalisation strategies are related through braid-like relations, as in Artin monoids, that have been used to propose a new method for a particular problem in homological algebra (namely, the construction of a contracting homotopy for the Koszul complex). The second result is published in Algebra and Representation Theory [20], the first one is submitted for publication [35], and both are contained in Cyrille Chenavier's PhD thesis [19].

Rewriting methods for coherence

Yves Guiraud and Philippe Malbos have written a survey on the use of rewriting methods in algebra, centered on a formulation of Squier's homotopical and homological theorems in the modern language of higher-dimensional categories. This article is intended as an introduction to the domain, mainly for graduate students, and will appear in Mathematical Structures in Computer Science [23].

Maxime Lucas, supervised by Yves Guiraud and Pierre-Louis Curien, has applied the rewriting techniques of Guiraud and Malbos [68] 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. This result is published in the Journal of Pure and Applied Algebra [25]. Maxime is currently engaged into a major rework of the results of [9], that will produce improved methods to build Squier's polygraphic resolution from a convergent presentation, based on the use of cubical higher categories instead of globular ones. He has already achieved a first result in this direction [77], and conducted a major foundational work towars the full result [78], which have just been submitted for publication.

Pierre-Louis Curien and Jovana Obradović pursued their work on cyclic operads (started in [36], now acepted in the Journal Applied Categorical Structures). They established the notion of categorified cyclic operad. Categorification involves weakening the axioms of cyclic operads (from equalities to natural isomorphisms) and formulating conditions concerning these isomorphisms which ensure coherence. For entries-only cyclic operads, this coherence is of the same kind as the coherence of symmetric monoidal categories: all diagrams made of associator and commutator isomorphisms are required to commute. However, in the setting of cyclic operads, where the existence of objects and morphisms depends on the shape of a fixed unrooted tree, these arrows do not always exist. In other words, the coherences that Mac Lane established for symmetric monoidal categories do not solve the coherence problem of categorified cyclic operads. They exhibited the appropriate conditions of this setting and proved the coherence theorem, relying on a result of Došen and Petrić, coming from the coherence of categorified operads. Additionally, by the equivalence between the two possible characterisations of cyclic operads, for cyclic operads introduced as operads with extra structure (that exchanges the output of an operation with one of its inputs), i.e. for exchangeable-output cyclic operads, they examined which of the axioms of the extra structure needs to be weakened (in order to lift that equivalence to weakened structures), and they exhibited the appropriate coherence conditions in this setting as well.