Which homotopy algebras come from transfer?

TL;DR

This paper characterizes when $A_ abla$-structures can be transferred over chain homotopy equivalences, introduces an obstruction theory for weak $A_ abla$-morphisms, and generalizes results to ${ m P}_ abla$-structures over fields of characteristic zero.

Contribution

It provides a complete characterization of transferability of $A_ abla$-structures and extends the theory to ${ m P}_ abla$-structures for quadratic Koszul operads.

Findings

Characterization of $A_ abla$-structures transferred over chain homotopy equivalences.

Development of an obstruction theory for weak $A_ abla$-morphisms.

Generalization to ${ m P}_ abla$-structures over fields of characteristic zero.

Abstract

We characterize $A_\infty$-structures that are transfers over a chain homotopy equivalence or a quasi-isomorphism, answering a question posed by D. Sullivan. Along the way, we present an obstruction theory for weak $A_\infty$-morphisms over an arbitrary commutative ring. We then generalize our results to ${\mathcal P}_\infty$-structures over a field of characteristic zero, for any quadratic Koszul operad ${\mathcal P}$.