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Looks like a good start, thanks! Nice diagrams!
I reworded from saying that a 2-functor is a functor to saying that it is a categorification of the notion of a functor, since functor focuses on the 1-categorical case. I also added some more whitespace between the last two xymatrix diagrams.
There is a notational inconsistency in the 2-functor page: sometimes, $\gamma_{f,g}$ is a natural transformation $P(g \circ f) \to P(g) \circ P(f)$, and other times it is $P(g) \circ P(f) \to P(g \circ f)$.
The latter seems correct, as it is the version depicted in the coherence diagrams, and matches the definition at pseudofunctor, but I’m not sure what else on the site depends on this definitional choice.
Please fix it.
It seems risky and redundant for this page and pseudofunctor to both contain explicit definitions of pseudofunctor. How about removing it from this page entirely and just pointing to pseudofunctor?
FWIW, the coherence diagrams on this page look nicer, but are only correct for strict 2-categories since they don’t include the associators or unitors. Also, the coherence axioms/diagrams for the unit constraint on this page still have it in the wrong direction; I didn’t fix them yet.
Edited to point out that the current pseudofunctor definition only works for pseudofunctors between strict $2$-categories, since the diagrams present are implicitly assuming that $1$-cell composition commutes on the nose. Will edit in the near future to add definition for pseudofunctor between bicategories.
Also edited to clarify naturality conditions on ’functor associators/unitors’.
have added (here) statement of the characterization of equivalences of 2-categories as the essentially surjective and fully faithful 2-functors.
Before recording this, I made a search through the literature listed at 2-category for both the words “fully faithful” as well as “full and faithful” and found no hits for a definition.
So I have added now pointer to
who make the definition explicit in their Def. 7.0.1 and state the characterization of 2-equivalences as Thm 7.4.1. These authors don’t quite say “fully faithful” for “equivalence on all hom-categories”, but close.
Correct definition of “lax”?
There seems to have been some mix-up in 2020, with point 6 switching the direction of lambda following a MO question about lax vs oplax, and then point 10 reverting for consistency with other bits but without switching the section about laxness.
I hope I have not got myself confused and messed it up further!
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