Note that surjective-on-objects equivalences are the acyclic fibrations in the canonical model structure on Cat.
]]>Yes, thanks, good point! The functor would have to be assumed to be surjective-on-objects, which is probably fine in practise in most cases. I’ll add something about this a little later to the nLab.
]]>That would only be true if you already knew there was an object such that on the nose.
]]>I guess this is fairly obvious actually. I’ll drop adjointness for simplicitly. Suppose that we have and , a natural isomorphism , and a natural isomorphism . Let be an object of . Since is an equivalence, there must be an isomorphism in such that is equal (on the nose) to the isomorphism . We can then replace by , where is the target of , and for is . Keeping the same, we then have that is on the nose equal to , and is still naturally isomorphic to .
If people agree, I’ll add this to equivalence or some such page. If I have not made a mistake in the above, it must be in the literature somewhere; does anybody know of an explicit reference? Or if there is some canonical/abstract formalism which recovers the above, that would be very good to add too.
]]>Given an (adjoint) equivalence in a 2-category, does anyone know if it is possible to replace the objects and/or 1-arrows in some reasonable way (up to equivalence/isomorphism) so that either the unit or the co-unit becomes an identity, not just a natural isomorphism? I don’t have time to think about it just now, and maybe someone knows something off the top of their head.
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