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This question is related to differential operator in noncommutative geometry. I wonder whether there is any approach to differential operator that taking differential operator as a functor? I think it might be easier to do localization from the functorial point of view.

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    $\begingroup$ A. Don't post two questions one immediately after the other. B. In commutative algebra, the Kahler differentials are defined by an adjunction, that is, it's more than functorial, it's a universal construction. Also, you should post some links (i.e., not links to the questions that you just posted) to relevant material, and also, you should add some motivation/work you've done yourself.. $\endgroup$ – Harry Gindi Jan 25 '10 at 9:11
  • $\begingroup$ Functor from what category to what category? $\endgroup$ – Yemon Choi Jan 25 '10 at 10:20
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    $\begingroup$ This is very vague; it seems outside the scope of what can usefully be asked on mathoverflow. There's been recent discussion of this at meta. $\endgroup$ – Scott Morrison Jan 26 '10 at 0:53
  • $\begingroup$ @YemonChoi Better late than never, I suppose, but yeah, the full-form of The adjunction is $A-\operatorname{Alg}/B \leftrightarrows B-\operatorname{Mod}$ (the first one is $A$-algebras with an $A$-algebra map to $B$) where $Y\mapsto \Omega_{Y/A} \otimes_Y B$ and its adjoint is given by $M\mapsto B\oplus M$. I happened to be looking back over things to find the precise adjunction, and it's given as proposition 1.7 of Quillen's Homology of Commutative Rings $\endgroup$ – Harry Gindi Nov 16 '17 at 8:44

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