Section: New Results

Homotopy of rewriting systems

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

The homotopical completion-reduction procedure

In  [39] , Stéphane Gaussent (Institut Camille Jordan), Yves Guiraud and Philippe Malbos have introduced the homotopical completion-reduction procedure as a higher-dimensional rewriting method to compute coherent presentations of monoids. The results of this procedure on Artin monoids of spherical type have been implemented by Yves Guiraud in a Python library, available on his webpage. The procedure is currently improved towards the explicit computation of full polygraphic resolutions of Artin monoids to provide a purely algebraic and constructive account of well-known geometric objects, such as Caylay graphs and Salvetti complexes.

In [16] , Yves Guiraud, Philippe Malbos and Samuel Mimram (CEA Saclay) have further investigated the homotopical completion-reduction procedure, extended with the adjunction/elimination of redundant generators, with successful application to two new classes of monoids: the plactic and the Chinese monoids. This work has been implemented by Samuel Mimram and Yves Guiraud into a prototype, that can be tested at http://www.pps.univ-paris-diderot.fr/~smimram/rewr , and has been presented to RTA 2013 by Philippe Malbos, where it has received the best paper award.

New methods for the computation of coherent presentations

During his M2 internship, Maxime Lucas, supervised by Yves Guiraud, has improved the rewriting method used in  [43] for the computation of homotopy bases of monoids and categories. This allows a more effective computation in several cases, based on the notion of Anick chain  [25] instead of the broader notion of critical branching. Maxime Lucas has now started a PhD thesis, supervised by Yves Guiraud and Pierre-Louis Curien, and currently investigates the use of Garside-like structures  [35] to further improve the computation of coherent presentations for higher-dimensional categories.

Higher-dimensional linear rewriting

Cyrille Chenavier, Pierre-Louis Curien, Yves Guiraud and Philippe Malbos investigate with Eric Hoffbeck (LAGA, Université Paris 13) and Samuel Mimram (CEA Saclay) the links between set-theoretic rewriting theory and the computational methods known in symbolic algebra, such as Gröbner bases  [28] . This interaction is supported by the Focal project of the IDEX Sorbonne Paris Cité. Yves Guiraud, Eric Hoffbeck and Philippe Malbos are currently working on an improvement, based on the homotopical completion-reduction procedure, of the methods known in algebra to compute homological invariants of algebras and operads. Cyrille Chenavier has started a PhD thesis, supervised by Yves Guiraud and Philippe Malbos, to use Berger's theory of reduction operators  [27] to design new methods for the study of rewriting systems.

Homotopical and homological finiteness conditions

Yves Guiraud and Philippe Malbos have written a comprehensive introduction [21] on the links between higher-dimensional rewriting, the homotopical finiteness condition “finite derivation type” and the homological finiteness condition “${\mathrm{FP}}_3$”, from the point of view of higher categories and polygraphs. The purpose of this work is to provide an introduction to the field, formulated in a contemporary language, and with new, more formal proofs of classical results.

In [19] , Yves Guiraud and Philippe Malbos have introduced a notion of identities among relations for higher categories presented by polygraphs. This notion is well-known in combinatorial group theory, where it is linked to the explicit computation of homological invariants and of formal representations of groups as crossed complexes. The main result of [19] is a procedure based on higher rewriting to compute generators of the identities among relations. They have related the facts that the natural system of identities among relations is finitely generated and that the higher category has finite derivation type (a homotopical finiteness condition introduced in  [43] for higher categories after Squier's work for monoids  [57] ).