Towards Koszulity for categorical structures: category of 2-nets

Abstract

Recently Batanin and Markl developed a theory of Koszul duality for operads over a general operadic category in the spirit of Ginzburg-Kapranov. However, their approach doesn't apply directly to the operadic category of n-trees, which are used to describe categorical structures. In this paper we present a category of (locally constant) 2-nets which addresses the main difficulty of extending their theory to the case of 2-trees. Specifically, our category of 2-nets is an operadic category containing 2-trees as a full subcategory, and every map between 2-nets admits a factorization into a chain of elementary maps.