Section: New Results

Effective higher-dimensional algebra

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

Higher linear rewriting

Yves Guiraud and Philippe Malbos have completed a four-year long collaboration with Eric Hoffbeck (LAGA, Univ. Paris 13), whose aim was to develop a theory of rewriting in associative algebras, with a view towards applications in homological algebra. They adapted the known notion of polygraph  [70] to higher-dimensional associative algebras, and used these objects to develop a rewriting theory on associative algebras that generalises the two major tools for computations in algebras: Gröbner bases  [69] and Poincaré-Birkhoff-Witth bases  [106]. Then, they transposed the construction of [12], based on an extension of Squier's theorem  [109] in higher dimensions, to compute small polygraphic resolutions of associative algebras from convergent presentations. Finally, this construction has been related to the Koszul homological property, yielding necessary or sufficient conditions for an algebra to be Koszul. The resulting work has just been submitted for publication [48].

Cyrille Chenavier has continued his work on reduction operators, a functional point of view on rewriting in associative algebras initiated by Berger  [63], on which his PhD thesis was focused [4]. First, using the lattice structure of the reduction operators, he gave a new algebraic characterisation of confluence, and developed a new algorithm for completion, based on an iterated use of the meet-operation of the lattice [29]. Then he related this completion procedure to Faugère's F4 completion procedure for noncommutative Gröbner bases  [80]. Finally, he gave a construction of a linear basis of the space of syzygies of a set of reduction operations, and used this work to optimise his completion procedure [46].

Cubical higher algebra

Maxime Lucas, supervised by Yves Guiraud and Pierre-Louis Curien, has applied the rewriting techniques of Guiraud and Malbos [93] 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 [32]. Then he has transposed to a cubical setting, and improved, the results of [12]. This first involved a deep foundational work on the connections between globular and cubical higher categories [52], generalising several already known links in a unique theoretical setting  [67], [68], [58], [111]. Then, he could prove Squier's theorem, giving a construction of a polygraphic resolution of monoids in the category of cubical Gray monoids [51]. All these results are contained in his PhD thesis, that was successfully defended in December 2017 [24].

Coherent Presentations of Monoidal categories

Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. Pierre-Louis Curien and Samuel Mimram have generalised this notion, in order to consider situations where the objects are considered modulo an equivalence relation, which is described by equational generators. When those form a convergent (abstract) rewriting system on objects, there are three very natural constructions that can be used to define the category which is described by the presentation: one consists in turning equational generators into identities (i.e. considering a quotient category), one consists in formally adding inverses to equational generators (i.e. localising the category), and one consists in restricting to objects which are normal forms. Under suitable coherence conditions on the presentation, the three constructions coincide, thus generalising celebrated results on presentations of groups. Those conditions are then extended to presentations of monoidal categories [30].

Categorified cyclic operads

The work of Pierre-Louis Curien and Jovana Obradović on categorified cyclic operads has been condtionally accepted in the Journal Applied Categorical Structures [47]. The revision will include a careful treatment of weakened identity laws, as well of weakened equivariance laws. It will also include the details of an example and an illustration of the work. The example involves a generalisation of profunctors, and the application is to the notion of anti-cyclic operad, which they prove to be “sign-coherent”.

Syntactic aspects of hypergraph polytopes

In collaboration with Jelena Ivanović, Pierre-Louis Curien and Jovana Obradović have introduced an inductively defined tree notation for all the faces of polytopes arising from a simplex by truncations, that allows them to view inclusion of faces as the process of contracting tree edges. This notation instantiates to the well-known notations for the faces of associahedra and permutohedra. Various authors have independently introduced combinatorial tools for describing such polytopes. In this work, the authors build on the particular approach developed by Došen and Petrić, who used the formalism of hypergraphs to describe the interval of polytopes from the simplex to the permutohedron. This interval was further stretched by Petrić to allow truncations of faces that are themselves obtained by truncations, and iteratively so. The notation applies to all these polytopes, and this fact is illustrated by showing that it instantiates to a notation for the faces of the permutohedron-based associahedra, that consists of parenthesised words with holes. In their work, Pierre-Louis Curien, Jovana Obradović and Jelena Ivanović also explore links between polytopes and categorified operads, as a follow-up of another work of Došen and Petrić, who had exhibited some families of hypergraph polytopes (associahedra, permutohedra, and hemiassociahedra) describing the coherences, and the coherences between coherences etc., arising by weakening sequential and parallel associativity of operadic composition. Their work is complemented with a criterion allowing to recover the information whether edges of these “operadic polytopes” come from sequential, or from parallel associativity. Alternative proofs for some of the original results of Došen and Petrić are also given. A paper containing this material has been accepted in the Journal Homotopy and Related Structure [33].

Opetopes

Opetopes are a formalisation of higher many-to-one operations leading to one of the approaches for defining weak $\omega$-categories. Opetopes were originally defined by Baez and Dolan. A reformulation (leading to a more carefully crafted definition) has been later provided by Batanin, Joyal, Kock and Mascari, based on the notion of polynomial functor. Pierre-Louis Curien has developped a corresponding syntax, which he presented at the workshop “Categories for homotopy and rewriting” (CIRM, September 2017).

Cédric Ho Thanh started his PhD work around opetopes in September 2017. His first contributions include a careful embedding of opetopic sets into polygraphs, and a (finite) critical pair lemma for opetopic sets. Indeed, opetopic sets seem to delimit a subset of polygraphs in which the basics of rewriting theory can be developped, without the anomalies already observed by Lafont and others happening, like the existence of a possibly infinite set of critical pairs in a rewriting system specified by finitely many rules. Opetopes are tree-like and hence first-order-term-like and that is the intuitive reason why these anomalies are avoided.

Higher Garside theory

Building on [9], Yves Guiraud is currently finishing with Matthieu Picantin (IRIF, Univ. Paris 7) a work that generalises already known constructions such as the bar resolution, several resolutions defined by Dehornoy and Lafont  [78], and the main results of Gaussent, Guiraud and Malbos on coherent presentations of Artin monoids [10], to monoids with a Garside family. This allows 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 with Matthieu Picantin, Yves Guiraud develops an improvement of the classical Knuth-Bendix completion procedure, called the KGB (for Knuth-Bendix-Garside) 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 [9]. The KGB completion procedure is partially implemented in the prototype Rewr, developed by Yves Guiraud and Samuel Mimram.

Foundations and formalisation of higher algebra

Yves Guiraud has started a collaboration with Marcelo Fiore (Univ. Cambridge) on the foundations of higher-dimensional categories, with the aim to define a general notion of polygraphs for various notions of algebraic structures. This is based on seeing higher categories as $n$-oids in a specific $n$-oidal category (a category with $n$ monoidal structures with exchange morphisms between them). With that point of view, a good notion of polygraph can be iteratively defined for monoids in any monoidal category with pullbacks, which is a sufficiently general setting for most purposes.

Eric Finster, Yves Guiraud and Matthieu Sozeau have started to explore the links between combinatorial higher algebra and homotopy type theory, two domains that describe computations with a homotopical point of view. Their first goal is to formalise the rewriting methods of [12] and [10] in homotopy type theory, establishing a first deep connection between the two fields. This direction will be explored further by Antoine Allioux, a PhD student co-directed by Guiraud and Sozeau, starting in February 2018.