This chapter presents rewriting techniques for associative algebras. Here, algorithms are sought to turn a given presentation by generators and relations into a rewriting system by orienting the latter, thereby producing linear bases of the presented algebra. In particular, this approach applies to various fundamental decision problems, such as the word problem, ideal membership, or to compute quadratic bases, e.g., Poincaré-Birkhoff-Witt bases, Hilbert series, syzygies of presentations, homology groups, and Poincaré series. However, if rewriting rules are required to be compatible with the linear structure, an immediate problem arises: no rewriting system can be terminating. In order to fix this problem, the structure of linear polygraph with an appropriate notion of reduction can be considered. Linear polygraphs are introduced as a framework for linear rewriting, their confluence properties are studied, and Gröbner bases and Poincaré-Birkhoff-Witt bases are expressed in the setting of linear polygraphs.

This chapter establishes 3-polygraphs as a notion of presentation for 2-categories. As expected, those consist in generators for 0-, 1- and 2-dimensional cells, together with relations between freely generated 2-cells, which are represented by generating 3-cells. Any 3-polygraph induces an abstract rewriting system, so that all associated general rewriting concepts make sense in this setting: confluence, termination, etc. However, more specific tools have to be adapted to this context: the notion of critical branching is defined here for 3-polygraphs, along with the proof that confluence of critical branchings implies the local confluence of the polygraph. In the case where the polygraph is terminating, local confluence implies confluence, providing a systematic method to show the convergence of a 3-polygraph. When this is the case, normal forms give canonical representatives for 2-cells modulo the congruence generated by 3-cells, and it is explained how to exploit this to show that a given 3-polygraph is a presentation of a given 2-category.

This appendix presents examples of coherent presentations of monoids. In particular, focus is placed on families of monoids which occur in algebra and whose coherent presentations are computed using the rewriting method that extends Squier’s and Knuth-Bendix’s completion procedures into a homotopical completion-reduction procedure. Coherent presentations of monoids are shown to explicitly describe the actions of monoids on small categories. This construction is applied to the case of Artin monoids. In particular, it is proven that the Zamolodchikov 3-generators extend the Artin presentation into a coherent presentation and, as a byproduct, a constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category is given. Coherent presentations of plactic and Chinese monoids are also provided.

This chapter introduces all the notions and tools necessary to define and establish the existence of the folk model category structure on ω-categories. Particular focus is placed on the concept of ω-equivalences, which will serve as the weak equivalences of this model structure. The class of ω-equivalences is the appropriate generalization to ω-categories of the class of equivalences of ordinary categories. In particular, an ω-equivalence between 1-categories is nothing but an equivalence of categories. To define this notion, it is necessary to generalize the concept of an invertible cell (or isomorphism). This leads to the notion of a reversible cell, which is, in intuitive terms, a cell admitting an inverse up to cells admitting inverses, up to cells admitting inverses, etc. Another fundamental tool is the ω-category of reversible cylinders in an ω-category, which will lead to a sensible notion of homotopy.

This appendix recalls basic notions on modules, abelian resolutions, and homology.

This chapter recasts the notion of string rewriting system into the language of polygraphs. This notion, which consists of a set of pairs of words called relations or rewriting rules over a fixed alphabet, is introduced along with a more general variant adapted to categories. It is shown that the rewriting paths form the morphisms of a sesquicategory, in which the traditional concepts for abstract rewriting systems can be instantiated. The word problem is then introduced, and it is shown that it can be efficiently solved for convergent, i.e., confluent and terminating rewriting systems. In practice, confluence can be checked by inspecting the critical branchings of the rewriting system, and termination by introducing a suitable reduction order. The convergence of a rewriting system is also useful to show that it forms a presentation of a given category. Finally, residuation techniques are introduced, which allow proving useful properties of categories (such as the existence of pushouts) by performing computations on their presentations.

This chapter establishes the main properties of the category of n-polygraphs. Limits and colimits are computed, and the category is proven to be complete and cocomplete. The behavior of the cartesian product deserves a special attention in that it does not correspond to the product of generators. The monomorphisms (resp. epimorphisms) in are then characterized as injective (resp. surjective) maps between generators. The linearization of polygraphic expressions plays a central role in proving these facts. Whereas the category of n-polygraph is a presheaf category in low dimensions, it already fails to be cartesian closed for n=3, the culprit for this defect being as usual the Eckmann-Hilton phenomenon. The categories of n-polygraph are, however, locally presentable. The technical notion of context is introduced in relation with n-dimensional rewriting, and used to prove that if an ω-category is freely generated by a polygraph, then this polygraph is unique up to isomorphism. Finally, rewriting properties of n-polygraphs are defined and coherence results are proven by rewriting on (n-1)-categories presented by convergent n-polygraphs.

This chapter is about proving the existence of the so-called folk model category structure on ω-categories, following Lafont, Métayer, and Worytkiewicz. This model category structure is a generalization of a model category structure on Cat whose weak equivalences are the equivalences of categories, a folklore result, hence the name. The analogous result for 2-categories was proved by Lack. The folk model category structure is a model category structure on ω-categories whose weak equivalences are the ω-equivalences and whose cofibrant resolutions are the polygraphic resolutions. It is the natural homotopical framework in which the notion of polygraphic resolutions lives. As convincing evidence of this, Métayer’s polygraphic homology can be expressed as a derived functor with respect to the folk model category structure.

Squier showed in his 1987 article that a convergent presentation of a monoid yields a partial resolution generated by the set of generators in dimension 1, by the set of rules in dimension 2, and by the critical branchings in dimension 3. If moreover the presentation is finite, the Squier resolution is finitely generated up to dimension 3. In this case, the monoid is said to be of homological type left-FP3. This property readily implies that the third integral homology group of the monoid is finitely generated. Therefore, a monoid whose third homology group is not finitely generated does not admit a finite convergent presentation. By explicitly exhibiting an example of this type, Squier first provided a negative answer to the question of universality of convergent rewriting.

This appendix recalls the main definitions associated to the notion of model category.

Anick and Green constructed the first explicit free resolutions for algebras from a presentation of relations by non-commutative Gröbner bases, which allow computing homological invariants, such as homology groups, Hilbert and Poincaré series of algebras presented by generators and relations given by a Gröbner basis. Similar methods for calculating free resolutions for monoids and algebras, inspired by string rewriting mechanisms, have been developed in numerous works. A purely polygraphic approach to the construction of these resolutions by rewriting has been developed using the notion of (ω,1)-polygraphic resolution, where the mechanism for proving the acyclicity of the resolution relies on the construction of a normalization strategy extended in all dimensions. The construction of polygraphic resolutions by rewriting has also been applied to the case of associative algebras and shuffle operads, introducing in each case a notion of polygraph adapted to the algebraic structure. This chapter demonstrates how to construct a polygraphic resolution of a category from a convergent presentation of that category, and how to deduce an abelian version of such a resolution.