posic | Из письма к Т.П.

<...> An algebra with m_0, m_1, and m_2 is just a CDG-algebra. By the way, algebras with m_0 should have m_1^2 not equal to 0, so their cohomology does not exist and it is not so obvious what should be meant by a quasi-isomorphism between them.

My own philosophy of the last 10 years or so is that algebras with m_0 are not the right object to consider. The good objects are:
1. A_\infty algebras (with m_1, m_2, m_3, ...)
2. CDG-coalgebras (with dual versions of m_0, m_1, and m_2)

In particular, even the de Rham CDG-algebra (corresponding to a vector bundle with a connection) should be considered as a coalgebra! I.e., the right object is the coring of polyvector fields over the ring of functions, with a certain structure on it corresponding to the de Rham differential on forms; this is called "a quasi-differential coring".

Furthermore, I actually know what is a quasi-isomorphism of conilpotent CDG-coalgebras. One considers a certain kind of good filtrations on coalgebras, for which the induced differential on the associated graded coalgebra always has zero square; so one can speak of filtered quasi-isomorphisms even though nonfiltered quasi-isomorphisms are not defined.

As to Koszul duality, there are two versions:
(1) Augmented (or non-unital) DG-algebras <---> conilpotent DG-coalgebras.
(2) Unital (but nonaugmented) DG-algebras <---> conilpotent CDG-coalgebras.
(Conilpotent coalgebras are always coaugmented, of course, or equivalently one can consider them as noncounital.)

For A_\infty-algebras, these dualities become just equivalences of categories:
(1) Augmented (or non-unital) A_\infty-algebras with A_\infty-morphisms between them = cofree DG-coalgebras.
(2) Strictly unital, with nonzero unit (but nonaugmented) A_\infty-algebras with strictly unital A_\infty-morphisms between them = cofree CDG-coalgebras.
Here "cofree" means cofree as a graded coalgebra (in the category of conilpotent graded coalgebras).

My favorite source on A_\infty-algebras is Lefevre-Hasegawa's dissertation (see Keller's homepage), which I have partly read just recently. Unfortunately, there is not a word about CDG-coalgebras there, he only knows DG-coalgebras.

<...> bar/cobar constructions and Koszul duality are the same thing. The only difference is that algebras sometimes have good generating subspaces, which have the Koszul property, but are significantly smaller than the whole algebra. In this case the Koszul dual coalgebra can be used as a replacement of the bar construction, being much smaller than the bar construction. The whole algebra is always a Koszul generating subspace of itself, and when you consider it as such, the Koszul dual object is just the bar construction.

I don't know if the notion of a Koszul generating subspace can be generalized to the A_\infty-algebras situation, except trivially by considering the whole A_\infty-algebra as the generating subspace.