Comparing notions of $A_{\infty}$ homotopy (in char 0): Markl's definition versus "Sullivan homotopy"

Question

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!

Answer 1

$\newcommand{\dd}{\mathrm d}$Markls notion is the same as an $A_\infty$-morphism from $A$ to $B\otimes C^\bullet_{[0,1]}$, where the second factor are the cellular cochains on the interval with (non-commutative!) dga structure defined via the Alexander-Whitney map. The two notions are then related by producing $A_\infty$-morphisms between this dga and $\Omega^\bullet_{[0,1]}$. At least the $A_\infty$-morphism from forms to singular cochains can be written down explicitly via so-called iterated integrals, compare this question I asked a while ago. There are still a lot of combinatorics involved, eg in finding the inverse $A_\infty$-morphism and taking tensor products of $A_\infty$-algebras.

Two recent articles which discuss generalisations of this question are Mazuir's Higher algebra of A∞ and ΩBAs-algebras in Morse theory II and Robert-Nicoud and Vallette's Higher Lie Theory.