Operads from posets and Koszul duality

Abstract

We introduce a functor $\mathsf{As}$ from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction $\mathsf{As}$ provides a generalization of the associative operad because all of its generating operations are associative. This construction has a very singular property: the operads obtained from $\mathsf{As}$ are almost never basic. Besides, the properties of the obtained operads, such as Koszulity, basicity, associative elements, realization, and dimensions, depend on combinatorial properties of the starting posets. Among others, we show that the property of being a forest for the Hasse diagram of the starting poset implies that the obtained operad is Koszul. Moreover, we show that the construction $\mathsf{As}$ restricted to a certain family of posets with Hasse diagrams satisfying some combinatorial properties is closed under Koszul duality.

40 pages

References in corpus (3)

• Combinatorial operads from monoids
• Enveloping operads and bicoloured noncrossing configurations
• Pluriassociative and polydendriform algebras

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• Cliff operads: a hierarchy of operads on words
• Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers
• Shuffle series
• Operads of decorated cliques I: Construction and quotients