Talk:Closed monoidal category

more common name
Maybe Monoidal closed category is the more common term? Geometry guy 21:01, 14 May 2007 (UTC)

Notation
Does anyone have a reference for the notation used in this article? I'm suspicious of the notation $$A\Leftarrow -$$ for the right adjoint to $$-\otimes A$$. In the case of Set this would make $$A\Leftarrow B$$ the set of functions from A to B!? It seems like $$-\Leftarrow A$$ would be better.

In any case, I would prefer to switch notation to that used in the reference by Kelly. He uses $$[A, -]$$ for the right adjoint to $$-\otimes A$$ and $$[\![A,-]\!]$$ for the right adjoint to $$A\otimes -$$. This notational appears to be slightly more common. It's also consistent with the notation used at closed category. Any thoughts? -- Fropuff (talk) 00:25, 15 February 2008 (UTC)


 * Your first question has an easy answer: I made a typo when writing that section, which I've fixed now. It's vastly more common to use lollipop-shaped symbols than to use $$\Leftarrow$$ or $$\Rightarrow$$, especially in linear logic, but I'm not sure how get those symbols here.


 * I like $$[A, B]$$ for the internal hom in symmetric or braided monoidal categories, and that notation is indeed very common. But I don't think that using $$[A,B]$$ versus $$[\![A,B]\!]$$ makes the right/left distinction clear, in the cases where that distinction really matters, and I don't think it's caught on. John Baez (talk) 16:46, 7 August 2009 (UTC)


 * Here: $$\multimap$$ its \multimap there is no \leftspoon or \rightspoon.


 * Well, I think that Fropuff is saying, in a roundabout manner, that one wants to write currying as
 * $$\text{Hom}(X, Y \Rightarrow Z) \simeq \text{Hom}(X\otimes Y, Z)$$
 * so that X,Y and Z are in sequential order on both sides, whereas the current left-right definition in this article mixes these up in a non-intuitive way. I'm trying to figure out if this is just definitional, or whether some knee-jerk sloppiness from symmetric or braided ideas has crept in.  Note that currently, both  and  writes these in the 'natural' sequential order. linas (talk) 13:01, 23 August 2012 (UTC)

OK, so the very first formula in the "definition" section is this:
 * $$B\mapsto A\otimes B$$

If instead, we write
 * $$B\mapsto B\otimes A$$

then for the third formula, we would get:
 * $$\mathbf{C}(B\otimes A, C)\cong\mathbf{C}(B,A\Rightarrow C)$$

right? This seems to me to be a better approach. To use a different notation, this mean the left-adjoint functor is
 * $$-\otimes A :\mathcal{C}\to\mathcal{C}$$

and the right adjoint functor is just as before:
 * $$[A, -]:\mathcal{C}\to\mathcal{C}$$

which is both what ncatlab says, and is what Fropuff says Kelley says, and I like that. So, yeah, I'm sort of gearing up to change the article to say this. Right? I'm not being stupid, am I? linas (talk) 03:30, 24 August 2012 (UTC)

Dammit. So, if I make the changes above, the first section looks pretty, and is in harmony with Kelley, with nlab, and with other WP articles. But then, the next section turns ugly: the current formula


 * $$\mathrm{eval}_{A,B} : A\otimes (A\Rightarrow B)\to B$$

which is pleasent, would have to be written as


 * $$\mathrm{eval}_{A,B} : (A\Rightarrow B)\otimes A\to B$$

which seems more awkward, somehow .... or does it? Hmmm... But I think harmonizing with everyone else is probably the most important thing to do. Its late, I'm going to bed.linas (talk) 03:56, 24 August 2012 (UTC)


 * OK I flipped everything around, correctly, I believe. linas (talk) 00:54, 25 August 2012 (UTC)

citation tag
I added the citation needed thing on the equivalent condition a while back. I think that in order for the second condition to imply the first, we have to assume that evaluation is natural in the second argument. -Sam W 67.170.52.2 (talk) 10:42, 27 March 2022 (UTC)