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I was in the middle of a long answer to this question, http://mathoverflow.net/questions/55721/categorification-request-closed, when it was unexpectedly shut down. I understand why it was closed, because it is indeed similar to some other MO questions (or has already effectively been largely addressed within an answer to another categorification request by Jan Weidner).
But my objection is that categorification of reciprocation, as in 1/1-x, is perhaps not a simple matter of rewriting the equation on both sides by 1-x and then categorifying that. For example, when a topologist writes a suggestive categorified equation like BG = 1//G, he surely doesn't mean that G x BG is homotopy equivalent to a point. So there is more to categorified reciprocation than simply reinterpreting it by multiplying both sides by the denominator. I was attempting to discuss such issues when the question was shut down.
Todd-
I understand your point, but I don't really agree with it; surely that topologist is saying that there is a fiber bundle over BG with fiber G which is contractible, which is a perfectly good categorification of the product. These questions are not precisely the same, but any good answer to one is a good answer to the other, and it's stupid to put them on separate pages. I strongly encourage you to answer Sammy's question instead of Jan's; I will happily vote it up.
Ricky: I've been tied up with other things and was only able to get back to the question this morning.
Sorry, Ben: I just posted before coming back to this meta discussion; I should have done it the other way around. "Categorification" is inherently a vague and slippery word: there is no one meaning of the term, so one could define categorified product as you are doing, but it's not the categorification that comes naturally to my mind; categorical or tensor product is. Anyway, I think what I wound up writing (at least insofar as I repeat the point I made above) is probably at this point a better fit to Jan's question, but we can continue talking about it if you disagree.
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