Quillen Homology for operads via Grobner bases
The main goal of this paper is to present a way to compute Quillen homology of a shuffle operad with a known Grobner basis. Similar to the strategy taken in a celebrated paper of David Anick, our approach goes in several steps. We define a combinatorial resolution for the ``monomial replacement'' of a shuffle operad, explain how to ``deform'' the differential to handle the general case, and find explicit representatives of Quillen homology for a large class of operads with monomial relations. We present various applications, including a new proof of Hoffbeck's PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operads of Batalin--Vilkovisky algebras and of Rota--Baxter algebras. The method of writing a resolution presented in this paper is very general. Namely, whenever you have a category which admits a theory of monomials (including Grobner bases and Buchberger algorithm) you can do the same procedure: First, take an object with a chosen Grobner basis. Second, define a resolution for an object with monomial relations (the monomial replacement of the starting object). Third, lower terms of relations will affect additional summands in the description of the differential in the resolution.