I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like:



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> 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.
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Another bloke remarked:



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> 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 $n$.
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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.



Homotopical interpretation of flatness?
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>).


