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Stream: theory: category theory

Topic: Notation for adjoints as an operation

Mike Shulman (May 04 2023 at 18:15):

Suppose $f$ is a morphism in a 2-category (or just a functor, or just an object of a non-symmetric monoidal category). What is a good pair of dual-looking notations for the left and right adjoints (duals) of $f$, if they exist?

I've recently been writing $f^*$ for the left adjoint and $f_*$ for the right adjoint, so that $f^* \dashv f \dashv f_*$, but this could be confusing since often $X^*$ just means "the" dual in a symmetric monoidal category, and in many contexts (e.g. geometric morphisms) we have $f^* \dashv f_*$ without an $f$ in the middle.

David Egolf (May 04 2023 at 18:56):

I don't know if you want a pre-existing notation, or if you're considering inventing a new one. So, I don't know what already exists - but how about $f^\vdash \dashv f \dashv f ^\dashv$? The left adjoint has the vertical bar on the left, and the right adjoint has the vertical bar on the right.

Mike Shulman (May 04 2023 at 19:02):

I don't think there's an existing "standard" notation, or at least if there is it's not standard enough for me to have heard about it. But I'd be interested both in notations that have been used already and proposals for new ones.

I can see $f^{\dashv}$ for the right adjoint, but I don't really like $f^{\vdash}$ for the left adjoint, because $\vdash$ is hardly ever used to denote adjunctions, and has other meanings such as a logical turnstile. Would ${}^{\dashv}f$ be too ugly?

Nathanael Arkor (May 04 2023 at 19:05):

Would ${}^{\dashv}f$ be too ugly?

It might be awkward if you're composing adjoints, because it may not be clear to which morphism the turnstile is associated without parentheses.

Nathanael Arkor (May 04 2023 at 19:06):

Another disadvantage of $f^* \dashv f_*$ is that some people have the opposite convention (e.g. in at least some of the literature on proarrow equipments).

Nathanael Arkor (May 04 2023 at 19:06):

(I don't have a suggestion, though; I'd also be interested in finding better notation.)

Matteo Capucci (he/him) (May 04 2023 at 19:07):

What about $f^! \dashv f \dashv f_!$?

David Egolf (May 04 2023 at 19:37):

For what it's worth, as a beginner to this stuff, I find moving notation up and down to indicate left/right very hard to remember. I'm sure this comes with practice, but I find it a bit arbitrary.

Reid Barton (May 04 2023 at 19:41):

I have used $f^L$ and $f^R$ (just for my own notes).

Reid Barton (May 04 2023 at 19:43):

In a [[pregroup grammar]] one usually uses $x^l$ and $x^r$ instead.

Reid Barton (May 04 2023 at 19:47):

Anything involving a subscript feels wrong to me because of the contravariance.

Graham Manuell (May 04 2023 at 19:53):

Nathanael Arkor said:

Another disadvantage of $f^* \dashv f_*$ is that some people have the opposite convention (e.g. in at least some of the literature on proarrow equipments).

This very upsetting.

Matteo Capucci (he/him) said:

What about $f^! \dashv f \dashv f_!$?

I'd have expected $f_!$ to be the left adjoint...

Mike Shulman (May 04 2023 at 21:23):

Ah, yes, I think I have seen $x^l$ and $x^r$. And I think I have seen $f^L$ and $f^R$ before too, perhaps in Lurie's work on [[fully dualizable objects]] in $(\infty,n)$-categories?

Mike Shulman (May 04 2023 at 21:26):

Hmm, I just glanced back at Lurie's paper and didn't see it, but I feel like I have seen this notation somewhere for 2-categories with all adjoints, $\dots \dashv f^{LL} \dashv f^L \dashv f \dashv f^R \dashv f^{RR} \dashv \dots$.

Mike Shulman (May 04 2023 at 21:26):

And it's a good point that subscripts tend to denote covariant operations and superscripts contravariant ones, and passage to both left and right adjoints is contravariant.

Reid Barton (May 04 2023 at 21:29):

That's the context in which I was using this notation so I would have believed I got it from Lurie's paper, if you hadn't said it's not in there.

Mike Shulman (May 04 2023 at 21:37):

I could have missed it.

Mike Shulman (May 04 2023 at 21:51):

In the notation of Fausk-Hu-May Isomorphisms between left and right adjoints, we have $f^* \dashv f_*$ and $f_! \dashv f^*$, while sometimes $f_* = f_!$ and other times $f^* = f^!$. So if we identified $f$ itself with $f_*$, we could have $f^* \dashv f \dashv f^!$.

On the other hand, Chaitanya pointed out to me that upper and lower stars and shrieks are all usually used for an induced action on presheaves/modules/etc., so it would be confusing to also use them for maps of domains. For instance, if $f:A\to B$ is a functor in $\rm Cat$, then $f^*$ often denotes the induced precomposition functor $PB \to PA$ on presheaves, so we wouldn't want to also use it for a left adjoint functor $B\to A$.

Mike Shulman (May 04 2023 at 21:52):

Another symbol that could be used for one or the other is $f^\dagger$. But I'm not sure which it would be, or what the other one would be. Maybe $f^\S \dashv f \dashv f^\dagger$?

Reid Barton (May 04 2023 at 21:58):

$f^\flat \dashv f \dashv f^\sharp$?

Mike Shulman (May 04 2023 at 21:59):

Tempting! But I'm doing this in the context of modal type theory where the morphisms of my 2-category represent modalities, and some of the modalities are actually themselves named $\flat$ and $\sharp$, so that would be kind of confusing...

Reid Barton (May 04 2023 at 22:03):

Yeah, that makes sense.

Kevin Arlin (May 04 2023 at 22:06):

Seems like $f^L$ is the only notation here that a reader can reliably guess the meaning of, at least once they know it's some adjoint to $f$ and maybe even before that. It's also a typographical generalization of $f^{-1},$ which seems good.

Reid Barton (May 04 2023 at 22:09):

I do generally associate this notation with the situation where every morphism has a left and right adjoint (which I assume is not the case for these modalities), but I don't see why that necessarily has to be the case.

Simon Willerton (May 05 2023 at 15:51):

Mike Shulman said:

Another symbol that could be used for one or the other is $f^\dagger$. But I'm not sure which it would be, or what the other one would be. Maybe $f^\S \dashv f \dashv f^\dagger$?

Just as is sometimes done with left and right duals in a monoidal category, you could do ${}^\dagger f \dashv f \dashv f^\dagger$.

Mike Shulman (May 05 2023 at 16:42):

I haven't seen that for duals in a monoidal category, do you have a reference?

Cole Comfort (May 05 2023 at 18:01):

Mike Shulman said:

I haven't seen that for duals in a monoidal category, do you have a reference?

Being his ex masters student, I know that Cockett uses this sort of notation for duals in non-symmetric linearly distributive categories (probably lots of his colleaguess as well). For example:

Cole Comfort (May 05 2023 at 18:01):

https://www.math.mcgill.ca/rags/bicats/bicat.pdf

Simon Willerton (May 05 2023 at 18:19):

I would associate it with quantum group people, they talk of left duality and right duality. I would suspect it's used in Kassel's book, but I don't have it to hand here. Shahn Majid's books would be another place to look. The first thing I could find on googling was this:

https://vainerman.users.lmno.cnrs.fr/enseign/M2QG.pdf

Simon Willerton (May 05 2023 at 18:22):

To be clear: they don't do this with $\dagger$, but with $\ast$.

Mike Shulman (May 05 2023 at 19:14):

Thanks. But it looks to me like the paper on linear bicategories uses $f^\perp$ (for the left adjoint) and $^\perp f$ (for the right adjoint)? (Page 14)

Cole Comfort (May 05 2023 at 19:23):

Mike Shulman said:

Thanks. But it looks to me like the paper on linear bicategories uses $f^\perp$ (for the left adjoint) and $^\perp f$ (for the right adjoint)? (Page 14)

Yes sorry for not being clear. I have never actually seen the dagger used, but I have seen the perp and star. edit: I didn't even notice that the adjoints were mixed up!

Leopold Schlicht (May 06 2023 at 10:03):

https://mathoverflow.net/a/351714

Mike Shulman (May 06 2023 at 14:26):

For the benefit of those who don't want to click on the link, someone asked this same question on MO three years ago, and was told that EGA writes $^{\rm ad}F \dashv F \dashv F^{\rm ad}$.

Mike Shulman (May 06 2023 at 14:26):

Thanks for the link! That looks kind of ugly to me though.

John Baez (May 06 2023 at 16:27):

I think "ad" could well be replaced by some symbol like $\dagger$ while preserving the good thing about this notation, namely that it's really easy to remember which is the left adjoint and which is the right adjoint.

Mike Shulman (May 08 2023 at 04:30):

Yes, I think ${}^\dagger f \dashv f \dashv f^\dagger$ is the best suggestion I've seen so far.

Nathanael Arkor (May 08 2023 at 07:26):

What's the typical method of disambiguating $f^\dag f$ with this notation? With $\circ$?

Mike Shulman (May 08 2023 at 13:48):

I don't know what's typical, but in my case, yes, I would use $\circ$.