Homotopy Transfer Theorem for linearly compatible di-algebras

Abstract

This paper studies the operad of linearly compatible di-algebras, denoted by \(As^{2}\), which is a nonsymmetric operad encoding the algebras with two binary operations that satisfy individual and sum associativity conditions. We also prove that the operad \(As^{2}\) is exactly the Koszul dual operad of the operad \(^{2}As\) encoding totally compatible di-algebras. We show that the operads \(As^{2}\) and \(^{2}As\) are Koszul by rewriting method. We make explicit the Homotopy Transfer Theorem for \(As^{2}\)-algebras.