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 ${\overline{\mathcal{M}}}_{0,n+1}$ 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.

37 pages; published by International Mathematics Research Notices, a few typos corrected

References in corpus (8)

• Operads, homotopy algebra and iterated integrals for double loop spaces
• Chow rings of toric varieties defined by atomic lattices
• Gröbner bases for operads
• The homology of real subspace arrangements
• A pairing between graphs and trees
• Toric varieties of Loday's associahedra and noncommutative cohomological field theories
• Word operads and admissible orderings
• Matroids, Feynman categories, and Koszul duality