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Perhaps the more interesting question is, given a braided 2-category and a braided category, how does one justify the claim that one is a categorification of the other?
This is a good family of examples, but it seems more algebraic than analytic. The construction with P yields a D-module supported on the zero set of P.
These are pretty good answers, and I guess a general intuitive picture is a tall order. I suppose one solution is to take an easy example and run through the literature while holding it in your head.
