Let $M$ be an object in an k-linear $k$-linear abelian category with enough projectives. Then one can construct an $A_\infty$-structure on the Ext algebra $$Ext^\bullet(M,M)$$ as follows: One chooses projective resolution $P\rightarrow M$ and forms the Hom complex $$Hom^\bullet (P,P)$$ Now the cohomology of this compex complex is the Ext algebra and in the case where k $k$ is a field, one can choose a "homotopy retraction" $Ext^\bullet(M,M)\rightarrow Hom^\bullet (P,P)$ and transfer the dg-algebra structure on $Hom^\bullet (P,P)$ along it.
My questions are:
- Why does this construction (up to $A_\infty$-isomorphism) not depend on a choice of projective resolution?
- One could try the same thing with an injective resolution, why is the result still the same?
