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• CommentRowNumber1.
• CommentAuthorKeith Harbaugh
• CommentTimeJan 14th 2020
Just checked the nLab page for bicategory, to learn nLab's views on the terminology for morphisms (of one type or another) between bicategories.
Unless I missed it, that page doesn't mention this topic, nor provide a link to a web page where it is discussed.
I presume you DO cover this topic at pseudofunctor (yes? no?).
Would it not be a good idea to provide a link from the bicategory page to where their morphisms are named and discussed?
BTW, what motivated the issue was: at one of my own web pages I am using the accepted term bifunctor for a functor of two variables.
For this non-expert, that motivated the question.
Finally, I could add such a link myself, but really think someone more familiar with how you like to style things should do it (obviously it's an easy edit).
• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJan 14th 2020

OK, sure, that makes sense. Thanks for bringing this to attention. I don’t plan on doing more than provide a link, so this won’t require much style.

Yes, pseudofunctor is one of the terms, as explicitly said at the linked article. As noted there, Bénabou used the term ’homomorphism’, as have others, but often, plain old ’functor’ is also used. I think ’bifunctor’ must be comparatively rare, so I think you’re safe there.

There is also a notion of lax morphism, which doesn’t assume invertibility of structural constraints (unlike the pseudo version). I believe morphism or functor without any extra qualifier generally defaults to the pseudo notion.

• CommentRowNumber3.
• CommentAuthorKeith Harbaugh
• CommentTimeJan 14th 2020
Thanks Todd; I also added pseudofunctor to the list of "Related concepts" for the bicategory page.

A bigger issue, which I'm sure has been discussed before at many places, is the lack of orthogonality in terminology.
One (at least I) would think that (bifunctor should be to functor) as (bicategory is to category).
What would be the problem with deprecating (bifunctor as a functor of two variables) in favor of making the above parallelism valid?
And lining up pseudofunctor with pseudocategory.
Is there a table somewhere showing how analogous definitions have (at the present) different terminology?
Probably such exists somewhere in Leinster's work, maybe it also somewhere here at nLab and I just don't know it.
If not, maybe when an expert has the time, creating such a page would be a good idea?
Especially since people can add comments to discuss the issues and problems.
• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJan 14th 2020

Keith, these are sociological questions. As far as I know, people simply don’t say ’bifunctor’ (at least not much, if at all) for this meaning, nor do they use ’pseudocategory’ to mean bicategory. In many cases, to do so would be to invite confusion (’biproduct’ is a good example: it has a meaning which is not the same as bi-product in a bicategory). This is discussed at bicategory, under Terminology.

Only rarely do we (at the nLab) play Humpty Dumpty with terminology. It’s a truism that language develops haphazardly, and for the most part our approach has been descriptive, recording language as it is actually used, and not necessarily as (some of us believe) it should be used. However, as you can see we do discuss from time to time notes on usage, in which user preferences or biases may creep in.

It’s true that the naming of $n$-categorical concepts has been especially haphazard (e.g., modifications between pseudonatural transformations, or perturbations between modifications). At some point one runs out of suitable English words and some systematization is in order. Thus, at some point we pushed for $(n, k)$-transfor as a useful general term.

If you look at pseudofunctor, you will read that many people simply say ’functor’ for a morphism between bicategories, and personally I think that’s fine, and part of a general trend towards systematizing use of terminology. I’m not in favor of trying to overturn decades of use of ’bifunctor’ for a functor with two arguments. In my opinion, there’s really no compelling need to do so.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJan 14th 2020

In my own writing, I long ago stopped using “bifunctor” in favor of “two-variable functor” for this reason (but I’m not about to go on any crusade about it). Usually one doesn’t even need a special word; it’s enough to say “functor $A\times B\to C$”.

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