## Section: New Results

### Effective higher-dimensional algebra

Participants : Antoine Allioux, Pierre-Louis Curien, Alen Durić, Eric Finster, Yves Guiraud, Amar Hadzihasanović, Cédric Ho Thanh, Matthieu Sozeau.

#### Rewriting methods in higher algebra

Yves Guiraud has completed a four-year collaboration with Eric Hoffbeck (Univ. Paris 13) and Philippe Malbos (Univ. Lyon 1), 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 [64] 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 [63] and Poincaré-Birkhoff-Witt bases [100]. Then, they transposed the construction of [14], based on an extension of Squier's theorem [104] 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 was published in Mathematische Zeitschrift [32].

Yves Guiraud has written and defended his “Habilitation à diriger des recherches” manuscript, as a survey on rewriting methods in algebra based on Squier theory [26]. The defense was held in June 2019.

Yves Guiraud works with Dimitri Ara (Univ. Aix-Marseille), Albert Burroni, Philippe Malbos (Univ. Lyon 1), François Métayer (Univ. Nanterre) and Samuel Mimram (École Polytechnique) on a reference book on the theory of polygraphs and higher-dimensional categories, and their applications in rewriting theory and homotopical algebra.

Yves Guiraud works with Marcelo Fiore (Univ. Cambridge) on the theoretical foundations of higher-dimensional algebra, in order to develop a common setting to develop rewriting methods for various algebraic structures at the same time. Practically, they aim at a definition of polygraphic resolutions of monoids in monoidal categories, based on the recent notion of $n$-oid in an $n$-oidal category. This theory will subsume the known cases of monoids and associative algebras, and encompass a wide range of objects, such as Lawvere theories (for term rewriting), operads (for Gröbner bases) or higher-order theories (for the $\lambda $-calculus).

Building on [9], Yves Guiraud is currently finishing with Matthieu Picantin (Univ. Paris Diderot) a work that generalises already known constructions such as the bar resolution, several resolutions defined by Dehornoy and Lafont [73], and the main results of Gaussent, Guiraud and Malbos on coherent presentations of Artin monoids [11], 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.

Yves Guiraud has started a collaboration with Najib Idrissi (IMJ-PRG, Univ. Paris Diderot) whose aim is to understand the relation between several different methods known to compute small resolutions of algebras and operads: those based on rewriting methods (Anick, Squier) and those that stem from Koszul duality theory.

#### Normalisation of monoids

Alen Durić started his Phd thesis (supervised by Yves Guiraud and Pierre-Louis Curien) in October 2019. His work so far has been mostly bibliographical. The goal is to combine methods from rewriting theory (and in particular the method of homotopical completion and reduction developed by Guiraud-Malbos-Mimram) and methods developed by Dehornoy and his coauthors in the study of monoids with Garside families, and by Dehornoy-Guiraud in the study of normalisation for monoids. Alen Durić is currently experimenting with some examples taken from these latter works, with the goal of building coherent presentations for them using the former methods.

#### Topological aspects of polygraphs

Amar Hadzihasanović joined the team at the end of November 2019, as a one-year postdoc funded by FSMP. He has been working intensively on the study of shapes appropriate for the description of higher cells as needed in various approaches to higher categories and higher structures. Amar Hadzihasanović's project is to recast his ideas in the framework of polygraphs, with the aim of bringing topological insights into the study of higher-dimensional rewriting.

#### Opetopes

The work of Pierre-Louis Curien, Cédric Ho Thanh and Samuel Mimram on syntactic and type-theoretic presentations of opetopes and opetopic sets has been submitted to a journal, and a short version has been presented at the LICS conference in Vancouver this year [45].

Cédric Ho Thanh, in collaboration with Chaitanya Leena Subramaniam, has defined the notion of “opetopic algebras” that leverages the subtle combinatorics of opetopes. This framework encompasses categories, planar operads, and Loday's combinads over planar trees. They have defined an opetopic nerve functor that fully embeds each category of opetopic algebras into the category of opetopic sets. In particular, they obtain fully faithful opetopic nerve functors for categories and for planar coloured operads. These results have been written up in [51]. This work is the first in a series aimed at using opetopic spaces as models for higher algebraic structures. In particular, the aim is to provide new models for infinity-categories and infinity-operads.

#### Foundations and formalisation of higher algebra

Antoine Allioux (PhD started in February 2018), Eric Finster, Yves Guiraud and Matthieu Sozeau are exploring the development of higher algebra in type theory. To formalise higher algebra, one needs a new source of coherent structure in type theory. During the first year of Allioux's PhD, they studied an internalisation of polynomial monads (of which opetopes and $\infty $-categories are instances) in type theory, which ought to provide such a coherent algebraic structure, inspired by the work of Kock et al [90]. They later realised that this internalisation is however incoherent as presented in pure type theory, essentially because of its reliance on equality types. Since then, they switched to a different view, describing opetopes as an external construction and relying on strict equalities in the metatheory to avoid the coherence problem. Opetopic type theory should then be, similarly to cubical type theory, a type theory indexed over these opetopic structures, where grafting and substitution are computional operations. They are now concentrating on showing that the modified inductive characterisation of opetopes and their algebras, still definable in type theory, gives rise to the standard notion of opetopes in mathematics, an original result in itself.