Unless I'm totally wrong, Cat is equivalent to the category of skeletal-categories-with cardinals, where by "skeletal" I mean "any two isomorphic objects are equal" and by "with cardinals" I mean over every object hovers a set that keeps track of the size of the isomorphism class you just collapsed. It really seems like this trick ought to be possible to continue, somehow, displaying the strict n-category "skeleton" of a weak n-category in a more obvious way, and keeping track of precisely the "error" in the approximation.
Unless I'm totally wrong, Cat is equivalent to the category of skeletal-categories-with cardinals, where by "skeletal" I mean "any two isomorphic objects are equal" and by "with cardinals" I mean over every object hovers a set that keeps track of the size of the isomorphism class you just collapsed. It really seems like this trick ought to be possible to continue, somehow, displaying the strict n-category "skeleton" of a weak n-category in a more obvious way, and keeping track of precisely the "error" in the approximation.
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