Operads from posets and Koszul duality

TL;DR

This paper introduces a functor from posets to operads, linking combinatorial structures to algebraic properties, and explores how these properties influence Koszulity and duality in the resulting operads.

Contribution

It establishes a new functor connecting posets to nonsymmetric operads, analyzing how combinatorial properties affect algebraic features like Koszulity and duality.

Findings

Posets with forest Hasse diagrams yield Koszul operads.

The functor ${ m As}$ generalizes associative operads from posets.

Operads from ${ m As}$ are rarely basic, depending on poset properties.

Abstract

We introduce a functor ${\sf As}$ 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 ${\sf As}$ 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 ${\sf As}$ 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 ${\sf As}$ restricted to a certain family of posets with Hasse diagrams satisfying some combinatorial properties is closed under Koszul duality.