# On the homotopy transfer of $A_\infty$ structures

## TL;DR

This paper investigates the transfer of $A_
fty$ structures, maps, and homotopies, providing detailed proofs and extending previous approaches by comparing with the homological perturbation lemma.

## Contribution

It offers rigorous proofs and extensions for the transfer of $A_
fty$ structures, clarifying and expanding upon earlier work in the field.

## Key findings

- Provided detailed proofs of $A_
abla$ transfer claims
- Extended previous methods by comparing with homological perturbation lemma
- Clarified the relationship between different transfer approaches

## Abstract

The present article is devoted to the study of transfers for $A_\infty$ structures, their maps and homotopies, as developed in \cite{Markl06}. In particular, we supply the proofs of claims formulated therein and provide their extension by comparing them with the former approach based on the homological perturbation lemma.