# Is the existence of $A_{\infty}$-inverse a consequence of Homotopy Transfer Theorem?

Let $$k$$ be a field of characteristic $$0$$ and $$(A,d_A)$$, $$(B,d_B)$$ be two differential graded (dg) algebras over $$k$$. Let $$f: A\to B$$ be a closed degree $$0$$ map of dg-algebras and $$g: B\to A$$ be a map of dg-vectors spaces such that $$gf-id_A\sim 0$$ and $$fg-id_B\sim 0$$.

My question is: could we extend $$g$$ to an $$A_{\infty}$$-map $$B\to A$$? Is it a consequence of the Homotopy Transfer Theorem?

Yes, see this paper of Martin Markl. Since $$B$$ is homotopy equivalent to an $$A_\infty$$-algebra, you can endow it with an $$A_\infty$$-structure and extend your maps to $$A_\infty$$-morphisms that yield a homotopy equivalence of $$A_\infty$$-algebras. The fact that $$f$$ is already a map of algebras and $$A$$ and $$B$$ are dg-algebras will simplify the formulas of the paper significantly.