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It appears that the notion of “left dual” and “right dual” in nLab differs from the literature. If we want “left dual” to coincide with “left adjoint”, shouldn’t we worry about the fact that $\otimes$ is the composition of 1-cells in the diagrammatic order? So, $i : I \to A^* \otimes A$ would be $i : I \to AA^*$ in the normal juxtaposition notation. That would mean that $A^*$ is the left adjoint of $A$. But, the page dualizable object calls it the “right dual” of $A$.
If we want “left dual” to coincide with “left adjoint”, shouldn’t we worry about the fact that $\otimes$ is the composition of 1-cells in the diagrammatic order?
It doesn’t have to be, does it? For example, think of composition of endofunctors as the monoidal product $\otimes = \circ$ on an endofunctor category $[C, C]$.
It seems to me that the stated convention is the adopted convention in much of the literature, although I’m feeling a little lazy to check sources right now.
Yes, if ${\otimes} = {\circ}$, then the terminology is consistent with the standard usage. Algebraists generally seem to use the diagrammatic order of $\otimes$, which becomes clear in its use for bimodules. (Even Ross Street seems to use it in “Quantum Groups: A path to current algebra”, at least on pages 21-22.) I didn’t realize nLab was using the opposite convention.
nLab was using the opposite convention.
There is really no such thing as “a convention on the $n$Lab”. Unlike in a text book by a single or a handful of authors, there is just no way that we can even try to enforce global conventions. Typically each author decides on his own. I guess we have quite a few pages where conventions change even within that one page.
Therefore the general rule is: whatever you write in some entry, try to make sure that you provide enough context that the relevant conventions are clear. If you find yourself writing “$\otimes$” for the first time on a page and are worrying if readers will read it as intended, be sure to add a paranthetical remark (where we use here the convention that…) or similar.
If you feel that you need to refer to a specific convnetion repeatedly over several nLab pages, just create another page that states the convention, and then simply point to that page.
One example of this that we have is the implicit infinity-category theory convention-page.
We’ve had long arguments about diagrammatic vs. Leibnizian order at the n-Category Café as well. Urs is right: we can’t even get agreement between ourselves on preferences (you will find for example multiple spellings of the same word on the same page).
I’m going to go into dualizable object and see whether a note needs to be made (as it probably does).
Edit: well, I went in, and the remark following the definition already looked pretty good to me. I added just a few extra words of amplification.
There is a very highly regarded category theorist who can’t seem to make up his mind which order he wants to use, and admits at the blackboard that he is bound to get confused if he tries to write something down. Me personally: I made a firm decision long ago that whatever advantages there seem to be with diagrammatic, I was going to stick with Leibniz. But since this can be confusing to others, I try to remember to stick in a $\circ$ symbol, to make clear my own convention. But I probably forget to do that sometimes.
Mike makes comments about this exact issue at the bottom of p.655 of Framed bicategories and monoidal fibrations.
BTW, I couldn't guess what the example Sup (notice that this link leads nowhere) was supposed to be, and there was no explanation. So I changed it to Ban, which contrasts nicely with Vect (as Rel contrasts with Set). Hilb might be even more familiar, but the context seemed to call for a category where not every object is dualisable.
I would imagine $Sup$ stood for the category of sup-lattices, the idea being that the canonical map from an infinite coproduct to an infinite product is an isomorphism, generalizing the biproduct case for $Vect$. I don’t think this is true for $Ban$ (and the only Banach spaces which are dualizable using the projective tensor product, the one used for the standard closed symmetric monoidal structure, are the finite-dimensional ones). Whereas for sup-lattices $X$, $Y$ in a topos $E$, we calculate
$(X \otimes Y)^\ast \cong \hom(X \otimes Y, \Omega^\ast) \cong \hom(X, \hom(Y, \Omega^\ast)) \cong \hom(X, Y^\ast)$(where $X^\ast$ for a complete lattice $X$ is the opposite $X^{op}$, also a complete lattice, and the subobject classifier $\Omega$ plays the role of the monoidal unit). We get then a canonical map
$\Omega \to (Y^\ast \otimes Y)^\ast$and at least in some cases such as power sets $Y = P X$, we have a self-duality $(Y^\ast \otimes Y)^\ast \cong Y^\ast \otimes Y$ which leads to $Y$ being dualizable. I’d like to think about this example more tomorrow after I get some sleep, but I’m pretty sure that’s what the example meant.
the only Banach spaces which are dualizable […] are the finite-dimensional ones
Ha, I was thinking of reflexive spaces, but of course that's not enough.
I agree with you.
As the person who originally wrote Sup, I can confirm that SupLat is what I meant.
However, returning to the topic of #1, I was under the impression that the meaning of “right dual” was fairly standard insofar as it refers to the order in which things are written on the page, and that this standardization was contrary to what the page dualizable object currently says: namely, that a right dual of $A$ is an object $A^*$ with an evaluation $A^* \otimes A \to I$ and a coevaluation $I to A\otimes A^*$. Am I wrong?
Am I wrong?
I don’t know! I really haven’t gone through the literature.
You want to change it? I actually don’t have a problem with that (although I feel as though I would privately tend toward the other convention).
I haven’treally gone through the literature myself either; I’m just going by what I’ve been told. And apparently I was the one who wrote the convention that’s been on the page for 3 years! I guess back then I hadn’t been told or internalized yet that the convention is as I claimed. I’ve changed it; but if anyone disagrees, feel free to object.
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