An Alternative to Homotopy Transfer for $A_\infty$-Algebras

TL;DR

This paper introduces a new method for constructing $A_ abla$-algebras from homotopy data, providing an alternative to the traditional homotopy transfer, and explores conditions for their equivalence or obstruction.

Contribution

It presents a novel approach to $A_ abla$-algebra transfer that can produce inequivalent structures and analyzes the conditions under which they are quasi-isomorphic or obstructed.

Findings

New $A_ abla$-algebras can be constructed from homotopy data.

Under certain conditions, these algebras are quasi-isomorphic to the homotopy transfer.

Obstructions exist preventing some $A_ abla$-algebras from being quasi-isomorphic.

Abstract

In this work, we propose a novel approach to the homotopy transfer procedure starting from a set of homotopy data such that the first differential complex is a differential graded module over the second one. We show that the module structure may be used to induce an $A_\infty$-algebra on the second differential complex, constructed in a similar fashion to the homotopy transfer $A_\infty$-algebra. We prove that, under certain conditions, the $A_\infty$-algebras obtained with this procedure are quasi-isomorphic to the homotopy transfer one. On the other hand, when the side conditions do not hold, we find that there are cases where the existence of an $A_\infty$-quasi-isomorphism with the homotopy transfer $A_\infty$-algebra is obstructed. In other words, we obtain a new $A_\infty$-algebra on the second complex, inequivalent to the homotopy transfer one. Lastly, we prove that these $A_\infty$-algebras are not infinitesimal Hochschild deformations of the homotopy transfer $A_\infty$-algebra.