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• CommentRowNumber1.
• CommentAuthorAlec Rhea
• CommentTimeFeb 25th 2019

Added actual definition for pseudofunctor and modified notions, moved discussion from idea section to new discussion section at bottom of page.

1. 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.

• CommentRowNumber3.
• CommentAuthorAlec Rhea
• CommentTimeFeb 26th 2019
• (edited Feb 26th 2019)

Fixed a labelling error in a diagram. (also thank you Richard!)

• CommentRowNumber4.
• CommentAuthorHurkyl
• CommentTimeMay 10th 2019
• (edited May 10th 2019)

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 11th 2019

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeNov 6th 2020

Fix mistake in the direction of the coherence cells for lax functors, as pointed out at https://mathoverflow.net/q/375699/49.

Also removed the term “associator” for the composition constraint, which doesn’t seem appropriate to me.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeNov 6th 2020

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?

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeNov 6th 2020

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.

• CommentRowNumber9.
• CommentAuthorAlec Rhea
• CommentTimeDec 7th 2020

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’.

• CommentRowNumber10.
• CommentAuthorAlec Rhea
• CommentTimeDec 7th 2020

Made data/diagrams consistent in direction throughout definition section.

• CommentRowNumber11.
• CommentAuthorAlec Rhea
• CommentTimeDec 7th 2020

Fixed direction of functor unitor diagrams, and realized why it makes more sense to have the functor associator point in the other direction. Everything should be consistent for now at least.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 28th 2021

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.

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