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I noticed that an entry bifunctor was still missing, though requested by some existing entries. So I briefly added something.
I linked some related concepts whose pages don’t exist but should. (I might make those pages tomorrow if I’m reminded, but I don’t have time now.)
I also added a Disambiguation section. I’ve put up two of these sections now, which I think are more visible than anything outside the contents, even though we’d expect people to ignore them the second time through the page. I’m undecided on whether it also needs a sentence like ‘This article is about functors of two variables.’ at the beginning.
Good points, thanks!
In similar ambiguous cases in several other entries we have a disambiguation line at the very top, even before the table of contents. Not sure how you feel about that, but I think it works well, as it seperates the disambiguation from the entry itself.
So I have re-edited bifunctor accordingly. Check out if you can live with that, I won’t be dogmatic about it, so let me know if you absolutely don’t like it. But if not, we should do something to various other entries where the same kind of thing happens.
Is there really anyone who uses “bifunctor” to mean “pseudofunctor”? I’m pretty sure I’ve never encountered that.
You may find it in some early work on ’bicategories”.
The pointer to “2-functor” will be sufficient in either case.
just to make things look nice I killed some grey links by creating what Toby was anticipating: binary function and binary morphism
In similar ambiguous cases in several other entries …
Yes, I’ve seen those (and written them a couple of times). But I wanted to try a different way and see how it worked.
At binary function should that be ’out of a product’ or ’out of a product of two objects’. There is otherwise ambiguity.
Okay, so you did like the different way better? Feel free to roll back my changes, in this case.
@Tim: fixed, and another point corrected (or made more specific).
@Urs: I did like it better, at least for now, partly because it allows for more discussion if needed. I put it back so people can decide if they like it, but if not, we can also put it back how you had it.
@Toby :-)
I did like it better, at least for now, partly because it allows for more discussion if needed. I put it back so people can decide if they like it, but if not, we can also put it back how you had it.
Okay. We should just try to agree on how to handle this globally.
I don’t think that we need everything to look the same. It’s good if people have a chance to see different things and decide how they like them. Even in the long run, our different ways of heading up the tables of contents doesn’t cause problems.
@Tim #5: Really? Benabou used “homomorphism” (and “morphism”) and nowadays I think category theorists generally say “pseudo functor” (or “lax functor”) or just “functor” if it is clear that the domain and codomain are bicategories. I’ve never seen “bifunctor” used for this, and I assumed it was because even to people with the (IMHO execrable) habit of using the prefix “bi-” to mean “bicategorical”, it was clear that “bifunctor” was already in common usage for a functor of two variables. I am happy with what the page bifunctor says now, but I’m curious: do you have references where it was actually used that way?
Also, I added links between bifunctor and two-variable adjunction.
Do people even say “bifunctor” a whole lot anymore? It sounds slightly old-fashioned to my ears, being merely a certain kind of functor. Sort of putting a hat on a horse.
Similarly, do people really say ’binary function’ much? Binary operation, sure, but ’binary function’? I can’t recall ever seeing it.
The real point of binary function, for me, is to make foundational comments that I would like to put there but haven’t. As for ‘bifunctor’, I thought that it meant functor between bicategories when I first saw it, but I don’t know if anybody has really used it.
Okay, that’s fine. I’ve still never seen ’binary function’ – my first thought was to contrast it with ’decimal function’. Can you give me a citation? (It’s not important if you can’t.)
’Bifunctor’ I’ve seen, in the sense of functor with two arguments. I’m not sure I’ve seen the bicategorical sense (it would have to be a distant memory).
I agree with Todd’s hat-horse comment. I don’t see the point of saying “bifunctor” for “functor whose domain is a product of two categories”. You can always just say “functor” instead (and I do).
It would be more justifiable if it were analogous to the situation with bilinearity. “Bilinear map $V \times W \to X$” means something different from “linear map $V \times W \to X$”. So we really do need the word “bilinear”. But a bifunctor $A \times B \to C$ is the same thing as a functor $A \times B \to C$. I don’t think we need the word “bifunctor” at all.
What would actually make more sense would be if category theorists used “bifunctor $A \times B \to C$” to mean “functor from the funny tensor product of $A$ and $B$ to $C$”. That would make a better analogy with “bilinear”. But I’m not advocating that!
I’ll join the others in saying that I don’t remember having heard “bifunctor” to mean some kind of map of bicategories.
The Encyclopedia of Mathematics and PlanetMath know “bifunctor” the way we have it in the entry, the references at Quillen bifunctor know it, too. For instance.
I remember that certainly In the math department of Hamburg people would speak of bifunctors. I did, too, complain that it’s just a functor out of a product, but if people say this word nevertheless, there is no harm in explaining what it means.
Of course there’s no harm! I’m just idly wondering if I’m right in thinking it’s slightly dated terminology. If it is, I wouldn’t mind inserting a note that says as much!
Ditto ditto for me about hats and horses. Note, though, that a Quillen bifunctor is not the same as a Quillen functor whose domain is a cartesian product. And it would be worth having the page bifunctor if for no other reason than to explain that it doesn’t mean functor between bicategories.
Can you give me a citation [ETA: for ‘binary function’]?
It’s used in computer science; otherwise, I’ve only seen ‘function of two variables’. If you Google it, you should find a Wikipedia page that was originally named by a computer scientist (even though the discussion is purely mathematical) and then a bunch of CS pages. I in fact learnt the term from the Wikipedia page, much of which is still what I wrote in it in 2003.
we really do need the word “bilinear”.
Maybe, maybe not. A bilinear map from $A$ and $B$ to $C$ may be thought of as a map from $A \times B$ to $C$, in which case it is not linear (except in degenerate cases); but it may also be thought of as a map from $A \otimes B$ to $C$, in which case it is linear. (Note that there is no such thing as a bilinear map from $A \times B$ to $C$ as such; you must specify the data $A$ and $B$, not just the datum $A \times B$. Of course, its clear what you mean when you write ‘$A \times B$’, but if $A$ and $B$ are themselves something complicated, then it’s not clear … although the usage of ‘$\oplus$’ instead of ‘$\times$’ can clarify.)
On the other hand, when working in an arbitrary multicategory (which may not have a tensor product), then we do need a word for a binary morphism … although we could just discuss a multimorphism from $A$ and $B$ to $C$ without the special term ‘binary morphism’.
I slightly reworded things at bifunctor, based on some of this discussion. I think what I wrote is factual.
The computer science context that I cited is for ‘binary function’, not for ‘bifunctor’. (It was @#19, first paragraph.)
I know, Toby – but I took it on myself to do a quick google on ’bifunctor’, and the first hits I saw involved some computer science context, so I went with that. (That could be a misleading impression, I suppose.)
OK. I was afraid that my comment (before I edited it) was rather ambiguous.
A bilinear map from $A$ and $B$ to $C$ may be thought of as a map from $A \times B$ to $C$, in which case it is not linear (except in degenerate cases); but it may also be thought of as a map from $A \otimes B$ to $C$, in which case it is linear.
Speaking of horses, that suggestion feels like it has the cart in the wrong place. Even if one prefers an explicit construction of $A\otimes B$ to a definition-by-universal-property, the reason we construct it that way is to make it be a representing object for bilinear maps!
Yes, just as the reason that we construct cartesian products is to represent functions of two variables as simply functions.
Just a propos, another example re
Do people even say “bifunctor” a whole lot anymore?
In
there is a planned section “7.5 Bifunctors” and the term appears throughout the text .
(Notice that I am not arguing any case about how terminology should be used. )
I generally avoid it, due to the potential confusion from the use of “bi-” to mean “co- and contra-” (e.g. biproduct) and also to mean “weak 2-” (e.g., um, “biproduct”?). It’s just as easy to say “a functor $C\times D \to E$”.
But I can see the point of it on that page, since it’s about to consider a weaker sort of two-variable functor.
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