I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like:
A map of commutative algebras is flat if pushing it out along any other morphism is quasi-isomorphic to the derived pushout when the algebras are embedded into dg-algebras. Hence flatness is just another name for quasi-fibrations.
Another bloke remarked:
Flatness is about giving the correct pullback, but it may also be obtained without flatness. Of course the correct condition is Tor-independence, i.e the tensor product $A\otimes_\Bbbk B$ is correct iff $\operatorname{Tor}^n_\Bbbk(A,B)=0$ for all $n$.
Where is this viewpoint written down and/or developed clearly? I'm not at a level where I can make out the picture from a few hints. Does 'derived' means 'homotopy' here? Why do dg-algebras pop up? What's this Tor-independence business all about?
Finally, the first bloke also wrote there's a homotopical characterization of étaleness, which I would also like to know more about.