In this question I will work over a field of char. $0$. Let $f,g\: : \: A\to B$ be dg algebras morphism between two cochain (commutative) dg algebras $A,B$ (assume positively graded). Let $h$ be a cochain homotopy between f and g, i.e. a linear map $h\: : \: A^{*}\to B^{*-1}$ such that $dh+hd=f-g$. This notion of homotopy seems to be not the correct one for dg algebras (see for example Do chain homotopic maps between dg-algebras induce “same” maps on dg-modules?). Let $\Lambda(t)$ be the dg algebra of polynomial forms on the interval $[0,1]$. The ''correct'' notion of homotopy between $f$ and $g$ seems to be: $f$ and $g$ are homotopic iff there exist a dg algebra morphism $H\: : \: A\to B\otimes \Lambda(t)$ such that $$ H(a,0)=f(a),\quad H(a,1)=g(a). $$ It is possible to turn $H$ in a cochain homotopy by define $h:=\int_{0}^{1}H$ such that: if $f$ and $g$ are homotopic via $H$ they are chain homotopic via $h$. Here my questions:

1) It seems to me that there should be an obstruction to produce an homotopy $H$ from a chain homotopy $h$ (see for example the მამუკაჯიბლაძე's comment in Do chain homotopic maps between dg-algebras induce “same” maps on dg-modules?). Is there a way to "measure" this obstruction?

2) Under which conditions on $A$ and $B$ does this obstruction vanishes?($A $ cofibrant? $A$, $B$ minimal algebras?)

3) The same discussion above works as well in the context of $A_{\infty}$ algebras. Assume that $A$ and $B$ are $A_{\infty}$ algebras. A notion of homotopy between $A_{\infty}$ algebra maps $f_{\bullet}, g_{\bullet}\: : \: A\to B$ may be the following: there exist an $A_{\infty}$ algebra map $H_{\bullet}\: :\: A\to B\otimes\Lambda(t)$ such that $$ H_{\bullet}(a,0)=f_{\bullet}(a),\quad H_{\bullet}(a,1)=g_{\bullet}(a). $$ Similarly we can define the notion of cochain homotopy between $A_{\infty}$ algebra maps. My question is: what will be the translation of question 1),2) in terms of $A_{\infty}$ ($C_{\infty}$) algebras?

4)Here a non trivial example. In this paper) about the homotopy transfer theorem an explicit chain homotopy between $A_{\infty}$ algebra maps is constructed. Let $A$ be a dg algebra and H(A) its cohomology. Consider a (cochain) quasi isomorphism $i\: : \: H(A)\to A$, a cochain map $p\: : \: A\to H(A) $ such that $pi=Id$. Assume that there exists a cochain homotopy between $ip$ and the identity maps on $A$. Then there exist

a) an $A_{\infty}$ structure on $H(A)$,

b) quasi isomorphism (of $A_{\infty}$ algebras) $g_{\bullet}$, $f_{\bullet}$ such that $f_{1}=p$ and $g_{1}=i$.

c) a "cochain homotopy*" $h_{\bullet}$ between $(g_{\bullet} \circ f_{\bullet})$ and the identity on $A$.

Here the question: Is it possible to turn $h_{\bullet}$ into $H_{\bullet}$ here? what will be the condition on $A$?

EDIT: As pointed out by Gabriel, the definition in Markl's paper is more elaborated. For $n>1$, $h_{n}$ is not merely a cochain map betwen $(f\circ g)_{n}$ and $0$.

problemwith your definition of homotopy which uses polynomial forms, but this is going to clash with just about any definition you find in the literature. Since $A_\infty$ works perfectly well in any characteristic and even over the integers, any reasonable reference will give a definition of $A_\infty$ homotopy which works in any characteristic and over the integers. Your definition using polynomial forms, which involves derivatives of polynomials and thus division by arbitrarily large integers, does not. $\endgroup$ – Gabriel C. Drummond-Cole Apr 20 '17 at 4:42