Homotopy Transfer Theorem for Linearly Compatible Di-algebras

TL;DR

This paper investigates the operad of linearly compatible di-algebras, establishing its Koszul duality with totally compatible di-algebras and providing an explicit Homotopy Transfer Theorem for these structures.

Contribution

It identifies the Koszul duality between linearly compatible and totally compatible di-algebras and explicitly formulates the Homotopy Transfer Theorem for $As^{2}$-algebras.

Findings

Proves $As^{2}$ is Koszul and dual to $^{2}As$.

Provides explicit Homotopy Transfer Theorem for $As^{2}$-algebras.

Rewrites operads to establish Koszulity.

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.