# Notation for “the” left adjoint functor

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).

• 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. – Adrien Feb 1 at 13:06
• @Adrien Thanks. Can you give an example of a book where it is used? – Martin Brandenburg Feb 1 at 15:59
• You could write it as a formula using limits: $d\mapsto \lim(\pi:d/F\to C)$. – Oscar Cunningham Feb 1 at 17:00
• 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. – Tim Porter Feb 1 at 17:19
• @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! – Martin Brandenburg Feb 1 at 20:32

In EGA 0.1.5.2-3 (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.
• 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. – Martin Brandenburg Feb 1 at 20:47
• I've seen the notation $\lceil f \rceil$ and $\lfloor f \rfloor$ being used for the isomorphism, mainly in a few comp sci papers – WorldSEnder Feb 1 at 23:53
• @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 $\mathbb{Z} \hookrightarrow \mathbb{R}$. – Robert Furber Feb 2 at 1:29