Posets from Cancellative Operads and Koszul Duality

Abstract

A species ${\mathcal{G}}$ is said to be homogeneous concentrated in m, if ${\mathcal{G}}[k]=\emptyset$ for $k\neq m$. Based on previous work of Fresse [Fre04], B. Vallette [Val07] proved that a quadratic cancellative operad generated by a homogeneous species is Koszul if and only if the maximal intervals of the associated posets P _Q are Cohen–Macaulay. In this chapter, we give an account of Vallete’s results and generalize his criterion for Koszulness. We show that the homogeneity assumption is not necessary. The result is still valid for any quadratic cancellative operad.