# Modules over quasiisomorphic DG algebras

Suppose there is a quasiisomorphism $$q: A \to B$$ between DG algebras. Is there some reasonable description of induced functor $$q^*: B-mod \to A-mod$$? Can we say something better if it was a $$B$$-equivalence, i. e. it induced quiso on bar constructions? (Or under any assumtions you wish to make.)

Some motivation behind this question: I believe that like in usual homotopy theory, in rational homotopy theory there ought to be some notions finer than weak equivalence — like collapsibilty or (more likely) simple homotopy equivalence. So, as an analogue of Whitehead torsion, we can try to define some obstruction class in $$H^2$$ of essential image with some coefficients describing extension of categories, but I'm afraid that it is wishful thinking — there's not much reason to expect $$q^*$$ to be a fibration.

• What sort of specific description are you looking for? The question as stated feels too vague to be answerable. – skd Oct 18 at 1:52