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.

This appendix recalls the definition of homology of a category with coefficients, as well as the notion of finite homological type for a category.

This chapter is about Métayer’s polygraphic homology of ω-categories. This homology theory was first defined in the following way: the polygraphic homology of an ω-category is the homology of the abelianization of any of its polygraphic replacements. Métayer then showed with Lafont that for every monoid, considered as an ω-category, its polygraphic homology coincides with its classical homology as a monoid. This result was then generalized to 1-categories by Guetta. In this chapter, it is proven that the polygraphic homology is the left derived functor of a linearization functor from the category of ω-categories to the category of chain complexes, respectively endowed with ω-equivalences and quasi-isomorphisms.

In this chapter, a notion of Tietze transformation for 2-polygraphs is introduced, consisting of elementary operations on 2-polygraphs that preserve the presented category, such that any two finite 2-polygraphs presenting the same category can be transformed into one another by applying such transformations. By using Tietze transformations, the goal is to turn a given presentation of a category into another one that possesses better computational properties. In particular, the Knuth-Bendix completion procedure applies those transformations to turn a presentation into a confluent one. Convergent presentations lead to a solution of the word problem: for those, the equivalence between two words is immediately decided by comparing their normal forms.To tackle the word problem for an arbitrary presentation, a good strategy consists in trying to transform it into a convergent one by using Tietze transformations. From this point of view, a natural question arises as to whether a finite presentation of a category with decidable word problem can always be turned into a convergent one by applying Tietze transformations: this problem is called "universality of convergent presentations".