**Question:** Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?

**Example:** Given a dg commutative algebra $A$, there is a dg Lie algebra $\operatorname{Der}(A)$ defined by understanding the notion of "derivation" internal to the category of dg vector spaces. If $A$ and $B$ are quasi-isomorphic dg commutative algebras, are $\operatorname{Der}(A)$ and $\operatorname{Der}(B)$ quasi-isomorphic dg Lie algebras? Note that any homomorphism $f: A \to B$ defines a (dg) vector space of "derivations relative to $f$", which is a bimodule for the Lie algebras $\operatorname{Der}(A)$ and $\operatorname{Der}(B)$ and receives maps as one-sided modules from each of these; one would expect these maps to be quasi-isomorphisms if $f$ is.

**Remark:** I intend my question to be somewhat open ended. As such, I would accept an answer that points me to the appropriate literature.