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 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.

We introduce the notion of species relative to a fixed hyperplane arrangement. Roughly speaking, a species is a family of vector spaces, one for each face of the arrangement, along with linear isomorphisms between vector spaces indexed by faces of the same support. Next, we introduce the notion of a monoid in species. It consists of a species equipped with "product'' maps from a vector space indexed by a face to a vector space indexed by a smaller face. These are subject to naturality, associativity, unitality axioms. There is also a dual notion of a comonoid in species defined using `"coproduct'' maps, and a mixed self-dual notion of a bimonoid in species. We also define commutativity for a monoid and dually cocommutativity for a comonoid. A bimonoid could be commutative, cocommutative, both or neither. Commutative monoids, cocommutative comonoids, bicommutative bimonoids are convenient to formulate using flats rather than faces. In addition to the above, we discuss related objects such as q-bimonoids (which include bimonoids, signed bimonoids, 0-bimonoids), signed commutative monoids, and partially commutative monoids. The latter interpolate between monoids and commutative monoids. The above notion of species when specialized to the braid arrangements relates to the classical notion of Joyal species.