There is a canonical and efficient way to extend a convergent presentation of a category by a 2-polygraph into a coherent one. Precisely, the 3-cells used in this extension procedure are in one-to-one correspondence with the confluence diagrams of critical branchings in the polygraph. Now, if the polygraph is finite, so is the set of its critical branchings, and therefore the set of 3-cells generating coherence can be taken to be finite. In such a situation, the polygraph is said to have finite derivation type, or FDT. The relevance of this concept, introduced by Squier, lies in the following invariance property: if a category admits a finite presentation having finite derivation type, then all finite presentations of also have FDT. This invariance will prove essential to show that some finitely presented categories do not admit convergent presentations. Using these conditions, Squier managed to produce an explicit example of a finitely presented monoid, with decidable word problem, but having no finite convergent presentation. This provides a negative answer to the question of universality of finite convergent rewriting.

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.