## Section: New Results

### Effective higher-dimensional algebra

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

#### Rewriting methods in algebra

Yves Guiraud has written with Philippe Malbos (Univ. Lyon 1) 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 has appeared in Mathematical Structures in Computer Science [32].

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 [71] 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 [70] and Poincaré-Birkhoff-Witt bases [107]. Then, they transposed the construction of [14], based on an extension of Squier's theorem [110] 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 will appear in Mathematische Zeitschrift [31].

Yves Guiraud has written his “Habilitation à diriger des recherches” manuscript, as a survey on rewriting methods in algebra based on Squier theory [13]. The defense is planned for Spring 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).

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, Cédric Ho Thanh and Samuel Mimram have developped (in several variants) a type-theoretical treatment of opetopes and finite opetopic sets, and have shown that the models of their type theory are indeed the opetopic sets as defined mathematically by the above authors. This work is being submitted to an international conference. Also, Cédric Ho Thanh has given a direct precise proof of the equivalence between many-to-one polygraphs and opetopic sets, thus establishing a connection with the theory of polygraphs [57].

#### Garside methods in algebra and rewriting

Building on [9], Yves Guiraud is currently finishing with Matthieu Picantin (Univ. Paris 7) a work that generalises already known constructions such as the bar resolution, several resolutions defined by Dehornoy and Lafont [79], 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

Antoine Allioux (PhD started in February), 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. Finster has developed 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 [96]. Antoine Allioux is focusing on building an equivalence of types between categories seen as polynomial monads and the standard univalent categories in Homotopy Type Theory [22]. Another result that should follow is the ability to define simplicial types in Homotopy Type Theory, a long standing open problem in the field. An article on this subject is in preparation. Once armed with such a definition mechanism for higher algebraic structures and their algebras, it should be possible to internalise results from higher rewriting theory in type theory, which was the initial goal of this project.

#### Type Theory and Higher Topos Theory

Eric Finster explored the connections between intensional type theory and the theory of higher topoi, as developed in the works on Joyal and Lurie [103]. In particular, in collaboration with Mathieu Anel, André Joyal and Georg Biedermann, he gave a proof of a new result about the generation of left exact modalities in higher topoi, which has a corresponding internalisation in Homotopy Type Theory. Applications of this result to the Goodwillie Calculus, an advanced technique in abstract homotopy theory, resulted in the article [28].