Philippe Malbos

Higher-dimensional normalisation strategies for acyclicity

We introduce acyclic polygraphs, a notion of complete categorical cellular model for (small) categories, containing generators, relations and higher-dimensional globular syzygies. We give a rewriting method to construct explicit acyclic polygraphs from convergent presentations. For that, we introduce higher-dimensional normalisation strategies, defined as homotopically coherent ways to relate each cell of a polygraph to its normal form, then we prove that acyclicity is equivalent to the existence of a normalisation strategy. Using acyclic polygraphs, we define a higher-dimensional homotopical finiteness condition for higher categories which extends Squiers finite derivation type for monoids. We relate this homotopical property to a new homological finiteness condition that we introduce here.

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 Squiers and Knuth-Bendixs completions into a homotopical completion-reduction, applied to Artins and Garsides presentations. The main result of the paper states that the so-called Tits-Zamolodchikov 3-cells extend Artins 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.

A Homotopical Completion Procedure with Applications to Coherence of Monoids

One 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.

Coherence in monoidal track categories

We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and use rewriting methods on polygraphs to solve it. The setting is extended to more general coherence problems, viewed as 3-dimensional word problems in a track category, including the case of braided monoidal categories.

Rewriting Systems and Hochschild-Mitchell Homology

Abstract This paper forms part of a project focusing on the development of homological and simplicial methods in rewriting. The purpose of this contribution is to generalize Kobayashis theorem for monoids to “monoids with several objects”. Following Squiers theorem, Kobayashi constructs a resolution for monoids presented by convergent rewriting systems. We construct a free acyclic resolution for kC, as a C-bimodule over a commutative ring k, where C is a small category provided with a convergent presentation. This resolution, associated to the Knuth-Bendix completion algorithm, reflects the combinatorial properties of C. In particular, categories admitting finite convergent presentations by graphs and relations have a finite type Hochschild-Mitchell homology.

Modeling Through Human-Computer Interactions and Mathematical Discourse

The project ϕ-calculus, or philosophical calculus, concerns the research on a rational framework that, in the context of interactive design, can be source and guarantee of the adequacy of a theory to its application context. The framework focus (i) on the process of theory construction by an Artificial Agent under the supervision of Human Agent(s), (ii) on the knowledge evolving through this process and (iii) on the adequacy of experiments with respect to knowledge represented in a given moment. The project is being developed in three different perspectives, namely, conceptual, formal, and experimental. Conceptual work consists on examining how experiments may point out the need for revising a theory and how to use experiments in order to perform revision. Formal work proposes formalisms to apprehend the dynamics on the process of theory construction, as well as building languages that may, from the one hand, facilitate interaction and, on the other hand, be adapted for computing. Experiments have been carried out in domains like e-commerce and Human Learning. In this paper, we focus on conceptual and formal aspects of the work on ϕ-calculus in its current state.

Polygraphs of finite derivation type

Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition left-FP3. Using this result, he constructed finitely presentable monoids with a decidable word problem, but that cannot be presented by finite convergent rewriting systems. Later, he introduced the condition of finite derivation type, which is a homotopical finiteness property on the presentation complex associated to a monoid presentation. He showed that this condition is an invariant of finite presentations and he gave a constructive way to prove this finiteness property based on the computation of the critical branchings: being of finite derivation type is a necessary condition for a finitely presented monoid to admit a finite convergent presentation. This survey presents Squiers results in the contemporary language of polygraphs and higher-dimensional categories, with new proofs and relations between them.

Knuth’s Coherent Presentations of Plactic Monoids of Type A

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth’s generators.

Homological Computations for Term Rewriting Systems

An important problem in universal algebra consists in finding presentations of algebraic theories by generators and relations, which are as small as possible. Exhibiting lower bounds on the number of those generators and relations for a given theory is a difficult task because it a priori requires considering all possible sets of generators for a theory and no general method exists. In this article, we explain how homological computations can provide such lower bounds, in a systematic way, and show how to actually compute those in the case where a presentation of the theory by a convergent rewriting system is known. We also introduce the notion of coherent presentation of a theory in order to consider finer homotopical invariants. In some aspects, this work generalizes, to term rewriting systems, Squier’s celebrated homological and homotopical invariants for string rewriting systems.