Suppose that we have $A_{\infty}$ algebras $A,B$ (over a field of characteristic $0$), with $A_{\infty}$ maps $f,g: A \rightarrow B$. In the paper https://arxiv.org/abs/math/0401007 Markl defines a notion of $A_{\infty}$ homotopy which looks like a some kind of elaborate chain homotopy condition (top of page 4, item $(6)$). At least over characteristic $0$, there is a competiting definition of a homotopy, defined as an $A_{\infty}$ morphism $H: A \rightarrow B \otimes \Omega^{\bullet}_{[0,1]}$ such that $H|_{t = 0} = f$ and $H|_{t =1} = g$.

Supposedly, in this context, the two definitions are equivalent. How can do I go from Markl's definition to the other definition?

Thank you for your time!

Suppose that we have $A_{\infty}$ algebras $A,B$ (over a field of characteristic $0$), with $A_{\infty}$ maps $f,g: A \rightarrow B$. In the paper https://arxiv.org/abs/math/0401007 (top of page 4, item $(6)$) Markl defines a notion of $A_{\infty}$ homotopy, which can analogously be used to define a homotopy between maps $f$ and $g$. At least over characteristic $0$, there is a competing definition of a homotopy, defined as an $A_{\infty}$ morphism $H: A \rightarrow B \otimes \Omega^{\bullet}_{[0,1]}$ such that $H|_{t = 0} = f$ and $H|_{t =1} = g$.

Supposedly, in this context, the two definitions are equivalent. How can do I go from Markl's definition to the other definition?

Thank you for your time!