1
$\begingroup$

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.

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ What sort of specific description are you looking for? The question as stated feels too vague to be answerable. $\endgroup$
    – skd
    Oct 18, 2019 at 1:52
  • $\begingroup$ One can take a two-dimensional non-abelian Lie algebra and map it onto a one-dimensional Lie algebra. This would induce a quasi-isomorphism of their Chevalley-Eilenberg coalgebras. So I'd strongly suspect that universal enveloping of this morphism will give a map of associative augmented algebras such that it is a bar-equivalence and not an isomorphism $\endgroup$
    – Grisha Papayanov
    Nov 21, 2019 at 9:57

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.