As far as I know, there is no "official" notation for the left adjoint of a functor $F : \mathcal{C} \to \mathcal{D}$ if it exists. I have seen the notation $F^*$ sometimes, but this looks only nice when $F$ is already written as $F_*$, which is not practical. (This notation is then motivated by direct and inverse image functors. And it seems to be quite common for adjunctions between preorders aka Galois connections.) Similarly, I have seen the notation $F_*$ for the right adjoint of $F$, which only looks nice when $F$ is already written as $F^*$. I have also seen the notation $F^{\dagger}$ for the right adjoint, which looks nice, but then how would you denote the left adjoint if it exists? Perhaps ${}^{\dagger} F$? I don't want to start a debate here what is a good notation or not, since this is subjective anyway and is not suited for mathoverflow. I would like to know:

Are there any textbooks, influential papers or monographs on category theory which have introduced a notation for the left adjoint of $F$? Is there any notation which has been used by multiple authors?

Just to avoid any misunderstanding: Of course there is the official notation $F \dashv G$ when $F$ is left adjoint to $G$, but $\dashv$ is a relation symbol. I am interested in a function symbol (which makes sense since left and right adjoints are unique up to canonical isomorphism if they exist).

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    $\begingroup$ I've no idea how widely used it is, but I like $F^L$ for the left adjoint and $F^R$ for the right one. $\endgroup$ – Adrien Feb 1 '20 at 13:06
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    $\begingroup$ @Adrien Thanks. Can you give an example of a book where it is used? $\endgroup$ – Martin Brandenburg Feb 1 '20 at 15:59
  • $\begingroup$ You could write it as a formula using limits: $d\mapsto \lim(\pi:d/F\to C)$. $\endgroup$ – Oscar Cunningham Feb 1 '20 at 17:00
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    $\begingroup$ I think the use of $F$ for a start is problematic. Often $F$ is the left adjoint as it is often used for 'free' with $U$ or similar standing for 'underlying'. Different contexts require different notation but I have used $L$ for the left adjoint and $R$ for the right when discussing adjoint functors when teaching category theory. It might help if you gave more idea of the context in which you are wanting this. $\endgroup$ – Tim Porter Feb 1 '20 at 17:19
  • $\begingroup$ @TimPorter I completely agree. But sometimes the letters $U,L,R$ are already taken, or when no other functor is in the context, it is very common to name the functor $F$. By the way, Borceux calls $F$ the right and $G$ the left adjoint, yikes! $\endgroup$ – Martin Brandenburg Feb 1 '20 at 20:32

In EGA (from the 1971 Springer edition) the right adjoint and the left adjoint of a functor $F$ are denoted by $F^{\rm ad}$ and ${}^{\rm ad}\!F$, respectively.

  • $\begingroup$ I am very happy about this answer, and I accept it right away (even though there might be other books with other notations) because EGA I has/had such a huge impact. I didn't read all of the preliminaries, so I wasn't aware that this notation is introduced there. Well spotted! $\endgroup$ – Martin Brandenburg Feb 1 '20 at 20:37
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    $\begingroup$ Let me also remark that the same section introduces a notation for the bijection $$\hom(F(x),y) \xrightarrow{\sim} \hom(x,F^{\text{ad}}(y)),$$ namely $f \mapsto f^{\flat}$, and $g \mapsto g^{\sharp}$ for the inverse. At least, this notation can be seen a lot in EGA. $\endgroup$ – Martin Brandenburg Feb 1 '20 at 20:47
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    $\begingroup$ I am quite sure that the "ad"-notation does not occur in EGA outside the cited section. $\endgroup$ – Fred Rohrer Feb 1 '20 at 21:11
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    $\begingroup$ I've seen the notation $\lceil f \rceil$ and $\lfloor f \rfloor$ being used for the isomorphism, mainly in a few comp sci papers $\endgroup$ – WorldSEnder Feb 1 '20 at 23:53
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    $\begingroup$ @WorldSEnder I think I've used this notation myself, and it comes by analogy to the fact that floor and ceiling are adjoints to the inclusion of posets $\mathbb{Z} \hookrightarrow \mathbb{R}$. $\endgroup$ – Robert Furber Feb 2 '20 at 1:29

P. Gabriel uses $F_\lambda$ for the left adjoint and $F_\rho$ for the right adjoint. See for instance p. 344 of the article "Covering spaces in representation theory" by K. Bongartz and P. Gabriel (Invent. Math. 65, 1982) (EuDML).


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