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.