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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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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.. – Harry Gindi Jan 25 '10 at 9:11
Functor from what category to what category? – Yemon Choi Jan 25 '10 at 10:20
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. – Scott Morrison Jan 26 '10 at 0:53

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