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John Baez (Dec 23 2020 at 20:03):What's a nice elementary reference that explains "normalized" pseudofunctors and proves that every pseudofunctor between 2-categories is equivalent to a normalized one? This is for the bibliography of a paper, so I want something official-looking that clearly explains this, if possible.
John Baez (Dec 23 2020 at 20:06):The fact I need actually follows from Proposition 4.1 here on the nLab: any pseudofunctor into Cat is pseudonaturally equivalent to a strict 2-functor. (I'm working with pseudofunctors to Cat).
John Baez (Dec 23 2020 at 20:07):However, I think there's some slightly different result that says any pseudofunctor is equivalent to a "normalized" one, meaning one that preserves identity 1-morphisms.
John Baez (Dec 23 2020 at 20:12):Johnson and Yau call this a "strictly unitary" pseudofunctor.
Ian Coley (Dec 23 2020 at 20:16):I was speaking with @Alexander Campbell about this recently -- I think you just force the structure isomorphism for the identity to be the identity and add some \phi_{id}^-1 to all the other structure isomorphisms. But I don't have a reference; he probably does.
John Baez (Dec 23 2020 at 20:25):Yes, that sounds right, but I don't want to talk about how it works in our paper - I just want to wave a magic wand, brandish a reference, and say "now it's normalized".
Alexander Campbell (Dec 23 2020 at 20:41):@John Baez See Proposition 5.2 of https://arxiv.org/abs/math/0607271
John Baez (Dec 23 2020 at 21:00):Hurrah! I was just looking at that paper, but somehow I only got to the part where they proved a similar "normalization" theorem for pseudonatural transformations.
John Baez (Dec 23 2020 at 21:01):Thanks for the Christmas present! :holiday_tree:
John Baez (Dec 23 2020 at 21:01):(I'm not Christian, but I still say stuff like that.)