Thomas wrote:
The latter makes sense in any 2-category whereas the former doesn't.
One can call a 1-arrow in a 2-category ff (following Street) if it is representably so. Only the essential surjectivity causes problems, as 'surjectivity' is in general relative to some subcanonical pretopology on the ambient category. In a paper in preparation I have shown that any 2-category with a class J of ff 1-arrows which form a (strict) subcanonical pretopology can be localised (as a bicategory) at J, and we recover all the usual theorems about internal categories, groupoids etc. No assumptions about limits or colimits past what is necessary for the definition of J, so there are applications to various 2-categories of structured Lie groupoids (poss. infinite dimensional) and so forth to nicer-behaved categories. (I presented some of this at the Australian Category Seminar in 2011.)
However, often you just get the "evil" version when not having a strong form of AC (for classes) available
Indeed, and then you call them 'weak equivalences', and find some sort of model structure, or just localise outright. Also, for the purposes of further discussion, the terminology 'evil' has been demoted to a mere footnote at the nLab, as it was probably always meant to be by its coiners, and replaced by the morally neutral and more informative name 'principle of equivalence'. Interestingly, Voevodsky's Univalence Axiom is a way of ensuring the principle is always respected (and without awkward acrobatics). Best regards, David Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]