TITLE: Homotopical interpretation of flatness?
QUESTION [9 upvotes]: 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.

REPLY [3 votes]: The derived tensor product is defined as a homotopy pushout of $M\leftarrow R\rightarrow N$ in commutative DGAs. Now, if $R$ is a commutative ring and $M$ and $N$ are $R$-algebras, then the homology groups $H^i(M\otimes^\mathbf{L}_R N)=\operatorname{Tor}^R_i(M,N)$. Therefore $M\otimes_R^\mathbf{L} N$ and $M\otimes_R N$ are quasi-isomorphic for all $N$ iff $M$ is flat. Tor-independence is an isomorphic condition (Definition 55.1 in http://stacks.math.columbia.edu/download/more-algebra.pdf).