This question is turning out to be a little long so let me start off with the headline. Given a differential graded algebra $A$, we can recover $A$ from its homology $HA$ if we know "the" $A_\infty$-structure of $HA$. Vaguely, I would like to know:
Question 0: Can this be done "functorially" in some sense?
Ok, now for the long version.
For a dg algebra $A$, the association $A\mapsto HA$ gives a functor from the category of dg algebras into the category of graded algebras. Kadeishvili's theorem states that there is a unique(ish) $A_\infty$-structure on the homology $HA$ with certain nice properties. In this way we can think of the association $A\mapsto HA$ as having values in the category of $A_\infty$-algebras. Unfortunately, there seems to be two problems when trying to make this into a functor:
- Given a dg morphism $f:A\to B$ the induced graded morphism $Hf:HA\to HB$ may not be an $A_\infty$-morphism
- One can get an $A_\infty$-morphism $f_*:=q\circ f\circ j:HA\to HB$ where $j:HA\to A$ and $q:B\to HB$ are $A_\infty$-quasi-isomorphisms, but then the association $f\mapsto f_*$ is no longer functorial.
This brings us to:
Question 1: Is there any way to fix this? Eg., can we somehow view homology as an $A_\infty$-functor or some other sort of "functor up to homotopy"?
Similarly, for a dg $A$-module $M$ there is an $A_\infty$-$HA$-module structure on $HM$ having nice properties.
Question 2: Can we recover the category of dg $A$-modules from the category of $A_\infty$-$HA$-modules, i.e., is there a functor $A$-mod$\to$ $HA$-$A_\infty$-mod (or better yet in the other direction) giving some sort of equivalence?
References also would be much appreciated.