# What notation should be used for right adjunct of arrow ($f^{\sharp}$ or $f^{\flat})$?

In the style of CWM let $$\mathcal A$$ and $$\mathcal X$$ denote categories and let there be an adjunction $$\langle F,G,\phi\rangle$$ from $$\mathcal X$$ to $$\mathcal A$$.

So for every pair $$(x,a)$$ there is a bijection:$$\phi_{x,a}:\mathcal A(Fx,a)\to\mathcal X(x,Ga)$$

In that situation I got accustomed to $$f^{\sharp}$$ as a notation for $$\phi_{x,a}(f)$$ which is also called the right adjunct of arrow $$f:Fx\to a$$.

For the notation of the inverse I got accustomed to $$f^{\flat}$$ as left adjunct for arrow $$f:x\to Ga$$.

I cannot recall anymore where I encountered these notations for the first time, and unfortunately I found on page 36 of this Introduction to Topos theory of Kostecky the same notation but then switched: $$f^{\sharp}$$ is used there for the left adjunct.

On page 147 of CWM where the Kleisly construction is handled it seems to be confirmed that $$f^{\flat}$$ should be used for the left adjunct (hence $$f^{\sharp}$$ for the right).

So MacLane seems to tell me that I am right, and Kostecki that I am wrong...

Off course notation on its own is not a big deal, but I do appreciate uniformity and will not persist in using my notation if it is somehow wrong.

Can someone tell me more about this notation and/or perhaps justify one of the two ways how to use it?

Which of the two ways is commonly used, and is there an underlying motivation for that?

• I also saw both in different books; I'm not sure there's an agreed upon convention – Max Feb 4 at 10:58
• Since in the generic case $f$ has only one adjunct, I see $\bar f$ as the most common notation, applying to either left or right as the context requires. – Kevin Carlson Feb 4 at 19:18

Another notation I've seen is $$\lceil f\rceil$$ and $$\lfloor g\rfloor$$, though it's not immediately obvious which way should be which. You could make a case for $$\lfloor-\rfloor$$ being the $$\mathsf{Hom}(FA,B)\to\mathsf{Hom}(A,UB)$$ direction by reference to the adjunction (Galois connection) for the embedding of integers into reals (which has left adjoint $$\lceil-\rceil$$ and right adjoint $$\lfloor-\rfloor$$). Using this same analogy vaguely argues for Kostecki's choice. I would be surprised if one couldn't make up a reasonable seeming analogy for the other direction, though.
• Thank you. If no other sounds appear (let's wait for a while, I postpone acceptance for now) then I will stick to my notation and have a good look whenever I meet it again somewhere else. Just like: what does the author means with $\subset$ (proper subset or subset)? – drhab Feb 4 at 11:23