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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeMay 22nd 2010
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   Author: TobyBartels
   Format: MarkdownItexTodd Trimble requested [[currying]] (on the Sandbox, of all places), and I wrote it (also linking to it from [[closed monoidal category]]).

   Todd Trimble requested currying (on the Sandbox, of all places), and I wrote it (also linking to it from closed monoidal category).

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeMay 22nd 2010
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   Author: TobyBartels
   Format: MarkdownItexThis also inspired me to write [[Baez's law]].

   This also inspired me to write Baez’s law.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeMay 23rd 2010
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   Author: Harry Gindi
   Format: MarkdownCould 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"?

   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"?

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeMay 23rd 2010
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   Author: TobyBartels
   Format: MarkdownItexI 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 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 […]’.)

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 23rd 2010
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   Author: Todd_Trimble
   Format: MarkdownItexI added some more material to [[currying]], bringing in particularly lambda calculus and cartesian closed categories, which predated the generalization to closed monoidal categories.

   I added some more material to currying, bringing in particularly lambda calculus and cartesian closed categories, which predated the generalization to closed monoidal categories.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeMay 23rd 2010
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   Author: TobyBartels
   Format: MarkdownItexRight, lambda calculus is where this idea originally came from.

   Right, lambda calculus is where this idea originally came from.

7. • CommentRowNumber7.
   • CommentAuthorHarry Gindi
   • CommentTimeMay 23rd 2010
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   Author: Harry Gindi
   Format: Markdown@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.

   @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.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 23rd 2010
   • (edited May 23rd 2010)
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   Author: Todd_Trimble
   Format: MarkdownItexI'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.

   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.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeMay 23rd 2010
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   Author: TobyBartels
   Format: MarkdownItexCurrying 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.)

   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.)

10. • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeMay 23rd 2010
    • (edited May 23rd 2010)
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexI 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 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.)

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 23rd 2010
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    Author: Mike Shulman
    Format: MarkdownItexI agree with Toby that currying usually refers only to the right-hand variable.

    I agree with Toby that currying usually refers only to the right-hand variable.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 23rd 2010
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    Author: Todd_Trimble
    Format: MarkdownItexIncidentally, it's Matthew 25:29-30. I've made the change at [[Baez's law]].

    Incidentally, it’s Matthew 25:29-30. I’ve made the change at Baez’s law.

13. • CommentRowNumber13.
    • CommentAuthorHarry Gindi
    • CommentTimeMay 23rd 2010
    • PermaLink
    Author: Harry Gindi
    Format: MarkdownItexWho exactly contrived to name it Baez's law?

    Who exactly contrived to name it Baez’s law?

14. • CommentRowNumber14.
    • CommentAuthorTobyBartels
    • CommentTimeMay 23rd 2010
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    Author: TobyBartels
    Format: MarkdownItex@ 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]].)

    @ 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.)

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeMay 23rd 2010
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    Author: Mike Shulman
    Format: MarkdownItexI 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 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.

16. • CommentRowNumber16.
    • CommentAuthorTobyBartels
    • CommentTimeMay 23rd 2010
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexI think that [[Baez's law]] is also a serious subject (terminology, the same as [[red herring principle]]), but the name is homorous.

    I think that Baez’s law is also a serious subject (terminology, the same as red herring principle), but the name is homorous.

17. • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeMay 24th 2010
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    Author: Mike Shulman
    Format: MarkdownItexFair 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."

    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.”