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Todd Trimble requested currying (on the Sandbox, of all places), and I wrote it (also linking to it from closed monoidal category).
This also inspired me to write Baez’s law.
Could you write something about using the word? If I want to present a map f(x,y):X x Y -> Z as a map f:X -> Hom(Y,Z), would I say that I am going to "curry the map through the first coordinate"?
I don’t know how you’d specify coordinates, although if I would be inclined to call your example currying through the second coordinate (with currying through the first coordinate making a map from $Y$ to $Hom(X,Z)$). But your example is the usual way, so one just says that one is going to ‘curry the map’. (Which I think that I said in the article: ‘we curry $f$ to produce […]’.)
I added some more material to currying, bringing in particularly lambda calculus and cartesian closed categories, which predated the generalization to closed monoidal categories.
Right, lambda calculus is where this idea originally came from.
@Todd: Would it be silly for us to just define what it means to curry a map in a certain coordinate? When your map f is simple, this is easy, but, for example, the currying that you do in the definition of the weighted colimit makes it a bit confusing if you don't specify what coordinate you're currying in.
I’m not sure why it’s confusing. If one said simply “curry $\mathbb{P}^{op} \times D \to Set$” without specifying which way it’s being curryed, then I guess it could be confusing. But since it’s specifically being curryed to $D \to Set^{\mathbb{P}^{op}}$, I don’t see how confusion is likely.
Toby, what do you think? Should we define currying in an argument?
I guess this is a good opportunity to ask about the terminology “closed monoidal category”. I think the tendency these days is for this to mean what used to be called “biclosed monoidal category”, and if there’s closure just on one side, then call it left closed or right closed. (So I guess if $- \otimes X$ has a right adjoint, you call it right closed; if $X \otimes -$ has a right adjoint, you call it left closed. It’s possible I have it backwards.) Am I correct about these recent tendencies, and if so should we update the page to reflect this?
Final question: my knowledge of the Bible is not so great that I knew offhand Matthew 5:29. Matthew 5 is the Sermon on the Mount – one of the most famous (and impressive) chapters anywhere in the Bible. But at the risk of seeming dense, I don’t really get why 5:29 is appropriate.
Currying in computer science seems to me to be quite specific: it goes from $X \otimes Y \to Z$ to $X \to [Y,Z]$, and that is it. It does not involve any other coordinate. If you want to go from $X \otimes Y \to Z$ to $Y \to [X,Z]$, then you need to make a call to the symmetry of $\otimes$ (if such a thing exists) yourself. These are computer scientists, after all; they don’t leave variations as an exercise to the reader, because their reader is the computer, and it is needs to be told explicitly. Perhaps it is just my limited experience in seeing it that way, but that’s how I always see it.
For reference, $\otimes$ associates on the left, so currying goes from $X \otimes Y \otimes Z \to W$ to $X \otimes Y \to [Z,W]$. Curry again to make that $X \to [Y,[Z,W]]$. (Again, if you want to associate on the right, then you need to put in the parentheses and the associator yourself. Note that the internal hom is often denoted with an arrow which associates on the right, so all is consistent.)
So currying happens in a right closed monoidal category (it is right, isn’t it?). If you have a left closed monoidal category, maybe you can call that ‘cocurrying’. (Well, that’s what I called in some Coq code I wrote once as a personal exercise.)
@ Todd re the Bible:
There is discussion (and quotation) in the reference listed at the bottom of Baez’s law. (And that is my only source for calling it the Matthew effect, although I like the name.)
I added a bit about currying through unusual or multiple variables. (I am pleased with my multiple layers of hats, which look nice in my browser with my fonts at least.)
I agree with Toby that currying usually refers only to the right-hand variable.
Incidentally, it’s Matthew 25:29-30. I’ve made the change at Baez’s law.
Who exactly contrived to name it Baez’s law?
@ Todd
Thanks for catching that! I don’t think that verse 30 is really relevant, which is why I left it out, but getting the chapter number wrong explains why you found it confusing!
@ Harry
I think John Baez. As a joke, so that it would be an example of itself. (I mean, he did think of it himself, but he never thought that he was first.)
Possibly the nLab needs a humor
category to put pages whose content, even titles, are not quite serious. So far, this might include Baez’s law, red herring, centipede mathematics, and negative thinking. (Also potentially abstract nonsense, although for now that redirects to category theory.)
I think that red herring principle, centipede mathematics, and negative thinking are serious pages about serious subjects, even though their names are a little bit of a joke, so I would be hesitant about putting them in a “humor” category.
I think that Baez’s law is also a serious subject (terminology, the same as red herring principle), but the name is homorous.
Fair enough. The point I was making is that I don’t think a category should be called “humor” if what it really means is “serious pages with humorous names.”
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