A Homotopical Completion Procedure with Applications to Coherence of Monoids

International audienceOne of the most used algorithm in rewriting theory is the Knuth-Bendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce an equivalent convergent rewriting system. It is in particular used to study presentations of monoids, since normal forms of the rewriting system provide canonical representatives of words modulo the congruence generated by the rules. Here, we are interested in extending this procedure in order to retrieve information about the low-dimensional homotopy properties of a monoid. We therefore consider the notion of coherent presentation, which is a generalization of rewriting systems that keeps track of the cells generated by confluence diagrams. We extend the Knuth-Bendix completion procedure to this setting, resulting in a homotopical completion procedure. It is based on a generalization of Tietze transformations, which are operations that can be iteratively applied to relate any two presentations of the same monoid. We also explain how these transformations can be used to remove useless generators, rules, or confluence diagrams in a coherent presentation, thus leading to a homotopical reduction procedure. Finally, we apply these techniques to the study of some examples coming from representation theory, to compute minimal coherent presentations for them: braid, plactic and Chinese monoids

Coherent presentations of Artin monoids

We compute coherent presentations of Artin monoids, that is presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squier's and Knuth-Bendix's completions into a homotopical completion-reduction, applied to Artin's and Garside's presentations. The main result of the paper states that the so-called Tits-Zamolodchikov 3-cells extend Artin's presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category

Coherent presentations of a class of monoids admitting a Garside family

This paper shows how to construct coherent presentations of a class of monoids, including left-cancellative noetherian monoids containing no nontrivial invertible element and admitting a Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of Deligne's original construction of coherent presentations for spherical Artin-Tits monoids: to general Artin-Tits monoids, and to Garside monoids. The result is applied to a dual braid monoid, and to some monoids which are neither Artin-Tits nor Garside. For the Artin-Tits monoid of type $\widetilde{A}_{2}$, a finite coherent presentation is given, having a finite Garside family as a generating set

Coherence of Gray Categories via Rewriting

Over the recent years, the theory of rewriting has been extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to low-dimensional weak categories, and consider in details the first non-trivial case: presentations of tricategories. By a general result, those are equivalent to the stricter Gray categories, for which we introduce a notion of rewriting system, as well as associated tools: critical pairs, termination orders, etc. We show that a finite rewriting system admits a finite number of critical pairs and, as a variant of Newman\u27s lemma in our context, that a convergent rewriting system is coherent, meaning that two parallel 3-cells are necessarily equal. This is illustrated on rewriting systems corresponding to various well-known structures in the context of Gray categories (monoids, adjunctions, Frobenius monoids). Finally, we discuss generalizations in arbitrary dimension

Polygraphs: From Rewriting to Higher Categories

Polygraphs are a higher-dimensional generalization of the notion of directed graph. Based on those as unifying concept, this monograph on polygraphs revisits the theory of rewriting in the context of strict higher categories, adopting the abstract point of view offered by homotopical algebra. The first half explores the theory of polygraphs in low dimensions and its applications to the computation of the coherence of algebraic structures. It is meant to be progressive, with little requirements on the background of the reader, apart from basic category theory, and is illustrated with algorithmic computations on algebraic structures. The second half introduces and studies the general notion of n-polygraph, dealing with the homotopy theory of those. It constructs the folk model structure on the category of strict higher categories and exhibits polygraphs as cofibrant objects. This allows extending to higher dimensional structures the coherence results developed in the first half

Coherent presentation for the hypoplactic monoid of rank n

In this thesis, we construct a coherent presentation for the hypoplactic monoid of rank n and characterize the confluence diagrams associated with it, then we use the theory of quasi-Kashiwara operators and quasi-crystal graphs to prove that all confluence diagrams can be obtained from those diagrams whose vertices are highest-weight words. To do so, we first give a complete rewriting system for the hypoplactic monoid of rank n, then, using an extension of the Knuth–Bendix completion procedure called the homotopical completion procedure, we compute the previously mentioned coherent presentation, which, from a viewpoint of Monoidal Category Theory, gives us a family of generators of the relations amongst the relations. These coherent presentations are used for representations of monoids and are particularly useful to describe actions of monoids on categories. The theoretical background is given without proof, since the main purpose of this thesis is to present new results

Topological Hochschild homology of Thom spectra and the free loop space

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space L(BX). This leads to explicit calculations of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p and HZ.Comment: 58 page