A Generalization of Operads Based on Subgraph Contractions

Abstract

We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $\operatorname{\overline{{\mathcal{M}}}}_{0,n+1}$, admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gröbner bases can be easily generalized to contractads. We verify the Koszul property of the commutative, Lie, associative, and Gerstenhaber contractads.