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• CommentRowNumber1.
• CommentAuthorAhmed
• CommentTimeSep 8th 2018

This is not a composition between functors but a composition between applications of the functors

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 1st 2019

1. Fixing left/right unitor names. Place left unitor first.

2. Fix left unitor name.

3. Update definitions of left/right unitors for consistency with other definitions (e.g., monoidal categories). Move left unitor before right unitor.

4. Update uses of left/right unitor.

5. Update whiskering for consistency re: recent changes to unitors.

6. Make ’such that’ clearer

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJan 13th 2020
Hi Ben,

Not sure why the links in the above messages don't render.
7. Update associator for consistency with monoidal categories and changes re: whiskering and left and right unitors.

• CommentRowNumber11.
• CommentAuthoratmacen
• CommentTimeJan 13th 2020

(This is a test. Here is Ben’s edit comment again.)

Update associator for consistency with monoidal categories and changes re: whiskering and left and right unitors.

8. I have fixed the edits now. Ben has no associated nForum user (the posts were wrongly associated initially with a banned existing user), hence why the posts now appear as made by the nLab edit announcer.

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeJan 14th 2020

• CommentRowNumber14.
• CommentAuthorKeith Harbaugh
• CommentTimeJan 14th 2020

Okay, I jumped in and added pseudofunctor to the list of “Related concepts”. I don’t think that hurts anything, and that is the first place I looked at originally.

• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeApr 26th 2020

Added the example of monoidal categories.

• CommentRowNumber16.
• CommentAuthorsamwinnick
• CommentTimeNov 9th 2021
• (edited Nov 9th 2021)
edit 4: Sorry about all the edits. I am back to what I thought originally, which is that I think there needs to be an additional axiom here. Namely, the horizontal composition bifunctor should be required to be associative, that is, horizontal composition of 2-morphisms should be associative. This associativity should be strict; in order to have a notion of weak associativity of horizontal composition of 2-morphisms, we would need to use 3-morphisms to form the pentagon identity and triangle identity.

The pentagon identity in the definition for weak 2-category a.k.a. bicategory has 2-morphisms as its arrows and 1-morphisms as its vertices, as one would expect since this 2-category is the oidification of monoidal category. Although the weak associativity decreed by the pentagon identity involves the horizontal composition bifunctor, it is about the (weak) associativity of the composition of 1-morphisms. This does not appear to imply any sort of associativity of the horizontal composition of 2-morphisms.

I think it would be perverse for horizontal composition of 2-morphisms not to be (strictly) associative. I only know of a few examples since I am only first learning this. One example is that the Godement product (of natural transformations) is (strictly) associative. This pertains to the 2-category Cat. But I think it should be required in general since the whole idea of the horizontal composition bifunctor is that it's supposed to work like composition in a 1-category.

Sam Winnick
• CommentRowNumber17.
• CommentAuthorsamwinnick
• CommentTimeNov 11th 2021
• (edited Nov 12th 2021)

To put it more explicitly, I suspect there should be one more axiom:

$(\alpha\circ\beta)\circ\gamma = \alpha\circ(\beta\circ\gamma)$

for all 0-morphisms $A$, $B$, $C$, $D$, 1-morphisms $A\stackrel{f_3}\to B\stackrel{f_2}\to C\stackrel{f_1}\to D$ and $A\stackrel{g_3}\to B\stackrel{g_2}\to C\stackrel{g_1}\to D$, and finally 2-morphisms $\alpha:f_1\to g_1$, $\beta:f_2\to g_2$, and $\gamma:f_3\to g_3$, where $\circ$ denotes horizontal composition.

This is not accounted for by the weak associativity axiom, since that pertains to the composition of 1-morphisms rather than of 2-morphisms.

I am a little confused about what exactly the coherence theorem for bicategories says, but wouldn’t horizontal associativity of 2-morphisms be something we would expect to have anyway? I can’t imagine not having to require it axiomatically.

hopefully the final edit: I had the details wrong again. Apologies to anyone whose time I wasted. The point stands: I think there should be this extra axiom.

• CommentRowNumber18.
• CommentAuthorTodd_Trimble
• CommentTimeJan 29th 2022

It’s not possible to edit or investigate edits, but comment 17 is mistaken, as one can see by considering identity 2-cells.

Thousands of people have looked over the definition of bicategory, so further edits that are not attested in the literature probably shouldn’t be performed, at least not without prior discussion.

• CommentRowNumber19.
• CommentAuthorDavidRoberts
• CommentTimeJan 29th 2022

Yes, not to pile on too much but the domains and codomains of the LHS and RHS of the proposed axiom don’t even match, if the composition of 1-arrows is not associative.

• CommentRowNumber20.
• CommentAuthorMike Shulman
• CommentTimeFeb 1st 2022
The correct version of this "axiom" is that these two composites of 2-cells agree "modulo" the associator isomorphisms. This is just the fact that the associator is a *natural* isomorphism, which is part of the standard definition.
• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeMay 9th 2022

I have touched the section “Coherence theorem” (here) adding more and more pertinent hyperlinks to technical terms.

Also cross-linked with Lack’s coherence theorem.