The purpose of this paper is to prove the koszulity of the operad ${ \mathcal Quad}$, governing quadri-algebras. That ${ \mathcal Quad}$ is Koszul was conjectured by Aguiar and Loday. The operad ${ \mathcal Dend}$, governing dendriform algebras, is known to be Koszul, by work of Loday, and ${ \mathcal Quad}$ is its second black square power. We find a new complex, based on the associahedron, which captures the structure of ${ \mathcal Dend}$. This complex behaves well with respect to the black squaring process, and allows us to conclude. Also, this proves koszulity of higher powers of ${ \mathcal Dend}$.
Categories: math.QA math.RA