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I agree with your convention, although there is really no chance of misunderstanding here since those arrows were thoughtfully labeled.
That ambiguity is the main reason I never use that sort of notation for adjunctions.
We are talking about this proposition I suppose?
Indeed the explicit “$\dashv$“-symbol is there to make clear which adoint is which.
It seems to me that the opposite of the convention stated in #1 is actually most common: people like to write the left adjoint on the left and the right adjoint on the right, as in
$L \colon C \leftrightarrow D \colon R \,.$Personally, in everything I ever type on the nLab I try to state the chirality explicitly via “$\dashv$” and in addition I try to follow the convention of displaying left adjoint arrows on top of their right adjoints.
For what it’s worth, I often write something like
$(L: C \to D) \; \dashv \; (R: D \to C).$@Todd: That is clearest as I often forget which way the $\dashv$ is supposed to go. :-)
Having learned topos theory quite soon after category theory, I’ve always stuck with the convention of #1.
A geometric morphism, of course, goes in the direction of its right adjoint, but for most adjunctions it doesn’t make sense to regard them as going in either direction. So I think it’s a bad idea to rely on any arbitrary convention (like that of #1) to convey information like which functor is which adjoint or which category is the domain of which functor. I generally use Urs’ convention which is not arbitrary (the left adjoint appears on the left), but Todd’s is even less potentially ambiguous because the placement of the arguments to $\vdash$ seem fortunately to be completely standard (with the exception of typos and ignorant mistakes). One could perhaps make it less redundant by omitting the type of one of the functors.
In case it was not clear, I meant that I write $L \dashv R : \mathcal{C} \to \mathcal{D}$ for Todd’s $(L : \mathcal{D} \to \mathcal{C}) \dashv (R : \mathcal{C} \to \mathcal{D})$. This corresponds to omitting the type of $L$.
@Zhen: I think $L \dashv R : \mathcal{C} \to \mathcal{D}$ is ambiguous unless you add parentheses $L \dashv (R : \mathcal{C} \to \mathcal{D})$; it might mean that you are regarding the entire adjunction as having an arbitrarily chosen “direction”, i.e. $(L \dashv R) : \mathcal{C} \to \mathcal{D}$.
#10 really makes sense to me in a way that no convention did before!
As Mike says, there are some missing parentheses, but leaving them out invites us to interpret things both ways (just as we leave out parentheses when using an associative operation). So, yes, the adjunction has a direction, but this direction is not arbitrary; it's the directions that makes the two parenthesizations equivalent. (And as a bonus, it agrees with the direction for geometric morphisms of toposes and continuous maps of locales.)
I agree with Mike in what he says in #12. I am not sure what Toby means in #13, mainly because the domains and codomains of the functors DID change between (#5,#6: $R : \mathcal{D} \to \mathcal{C}$ ) and (#10,#12: $R : \mathcal{C} \to \mathcal{D}$) :(
Yes, things changed from #6 to the alleged quotation of #6 in #10. In particular, $C$ became $\mathcal{D}$, and $D$ became $\mathcal{C}$. But doing that substitution, we know what is meant. (It helps that the font changed at the same time that the letters changed, so you can think $C = \mathcal{D}$ and $D = \mathcal{C}$ if you wish.)
Surely $\alpha$-equivalence is not a problem for us?
Re #13:
I thought about justifying it that way, but I think it’s actually in the same vein as writing “$A \implies B \implies C$” for “$A \implies B$ and $B \implies C$”: after all, when I write “$L \dashv R : \mathcal{C} \to \mathcal{D}$”, I am making two assertions: $L \dashv R$ and $R : \mathcal{C} \to \mathcal{D}$. A crude but convenient abbreviation.
I still think parentheses are needed. I think a direction is still arbitrary if it’s chosen in order to make a certain notation work out.
I don't know about you (Mike), but I choose conventions all the time to make notations work out. This is how we decide what's a left or right module, for example.
Since Paul Taylor recently accused me on Math Overflow of inappropriately treating Wikipedia as an authoritative reference, let me compound the crime: At the English Wikipedia's discussion of the composition of adjunctions, they agree with the convention advocated by various people here: that an adjunction goes in the direction of the right adjoint.
I put no trust in Wikipedia: it’s just more evidence that there are some people who think that that convention is standard, as we’ve already seen in this thread. (In case anyone needs an example of a situation in which it’s natural to treat an adjunction as going in the direction of the left adjoint, in a proarrow equipment every arrow gives rise to an adjunction of proarrows whose left adjoint points in the direction of the original arrow.)
We call a module “left” or “right” because we write its action on the left or right respectively, or one might say the reverse that we use that notation because of the name. But in that case the name literally indicates the notation to be used. An analogous thing for the direction of adjunctions would be something like saying a “left adjunction” or “right adjunction” to indicate which direction we consider it as pointing in.
It’s also true that when defining a new notion, one often chooses conventions to make the notations convenient. But there’s no way to impose a choice of the directionality of adjunctions on the world at this point.
Personally, I don’t think it’s very natural to consider an arbitrary adjunction as having a direction at all. We have to do it sometimes, but it’s always arbitrary. In some cases, there are reasons to prefer one choice or the other, but there’s no sense in trying to make or enfore a global convention across mathematics.
No, not a global convention, just a convention when using the notation $L \dashv R\colon D \to C$. If you want to go the other way, after all, you are always free to do so: $R \vdash L\colon C \to D$. The former is better for geometric morphisms, that's all.
I still maintain that that notation without parentheses can be confusing to someone who hasn’t been informed of the convention, or even who was informed of the convention 20 pages ago at the beginning of the paper. What’s wrong with a few parentheses?
I find the parentheses a very good compromise and will adopt them in future writings.
Re #8, for what it’s worth, I know a famous topos theorist who writes all geometric morphisms in the direction of the right adjoint but writes most other adjunctions in the direction of the left adjoint (so that the symbol for the left adjoint functor is on the left).
What’s wrong with a few parentheses?
That depends. $(L \dashv R)\colon D \to C$ would be terrible, because the reader needs to know what convention is used for the direction of an adjunction to parse that, and this is exactly the sort of convention that we can't expect the reader to know. $L \dashv (R\colon D \to C)$ would be fine, but at some point you might get lazy and start leaving the parentheses out.
What I think is most important is that, if you do decide on a convention for the direction of an adjunction (say to start composing them, to draw an analogy with continuous maps, etc), then it ought to match what you have been writing before. Then when you introduce this convention (as you must if you want to use it, and we are supposing here that you need to use it) you can say that you are reparenthesizing $L \dashv R\colon D \to C$, and it will make sense.
While I find it natural to fix a direction of a geometric morphism, as t comes from the geometric meaning of this notion, I find it completely unnatural assuming any direction for a general adjunction. For example, it is aggressive against the theory of cospaces which are often useful.
In old times people also talked about adjoint functors and coadjoint functors and,as it is with unmotivated conventions, now nobody remembers which one is left and which one is right adjoint to the given functor.
Toby: $L \dashv R\colon D \to C$ would mean to me by a guess that $L:D\to C$ and that $L\dashv R$ as $R$ is appended so I would attach the $D\to C$ to the first item from the side from which I read. Thus I would guess totally opposite from what is intended.
I never proposed $(L \dashv R)\colon D \to C$; I agree that that is terrible. And I would naturally make the same guess about $L \dashv R\colon D \to C$ as Zoran.
Why is it $\dashv R$ that is appended to $L$, and not $L \dashv$ that is prepended to $R$?
Based on Zoran’s and Mike’s natural guess, that supports using parentheses in $L \dashv (R\colon D \to C)$. Still, I expect that if I used that a lot, then I'd want to drop the parentheses eventually.
30: you are saying that knowing your usage history will tend to break the primordial symmetry and warrant dropping the paranthesis in the context of your histories. Of course… :)
Re #31: no, I think it just means he’d get tired of putting in the parentheses every time, which I find understandable. I wouldn’t mind his dropping them, so long as there is a convention stated on the nLab page in question.
For what it's worth, I'm not sure how often I would have any call to really do this. For a one-shot adjunction, it's best to just label everything, which is what is currently being done on the page that started this discussion; and even though I now feel like the arrows ought to be reversed there, what is written is perfectly clear, and so I wouldn't touch it.
On the other hand, if I'm deep into discussing geometric morphisms or something like that, then I probably would use a single letter (say $f$) for the adjunction, then use $f^*$ and $f_*$ for (respectively) the left and right adjoints. (See for example the notation at inverse image and direct image.) So the individual functors would have to come first and then recognized afterward as forming an adjunction, before I think that I would use that notation.
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