Note that in the above construction we made a number of choices; here we must beware. Choosing a good categorification – like designing a good algebraic structure such as that of preorders or quantales – is part of the art of mathematics. There is no prescribed way to categorify, and the success of a chosen categorification is rather empirical: its richer structure should allow us more insights into the subject we want to model.
-Brendan Fong
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