There are various naturally occuring chains of adjoint functors, e.g. between the category of simple graphs (with suitable class of morphisms) and Set, the functor $\pi_0$, sending a graph to its set of connected components, is left adjoint to the functor that forms the discrete graph on a set. This functor in turn is left adjoint to the forgetful functor, which itself is left adjoint to the functor that forms the complete graph on a set. This chain of length four cannot be continued on either side, but there are constructions of such chains of any finite length. (However, I do not know if there is a non-trivial (one- or two-sided) infinite chain.)
My question now is: Is it possible that such a chain forms a non-trivial loop? (By non-trivial, I mean a chain that does not come from either an adjoint equivalence or a self-adjoint functor.) For concreteness' sake, is there a nontrivial adjoint "triangle", i.e. three endofunctors $F,G,H$ on a category $\mathcal{C}$ such that $F$ is left adjoint to $G$, which is left adjoint to $H$, which again is left adjoint to $F$, without any two of these functors being isomorphic?
Thanks in advance for any information on this matter.
Yes. The simplest instance of this, in which we have $F \dashv G$ and $G \dashv F$, is called an ambidextrous adjunction and there are many examples (for instance, coincident limits and colimits).
The general question is tackled in Booth's Sequences of adjoint functors, in which he proves that cyclic adjoint changes of length $n$ exists for every positive integer $n$.
For a family of examples that arise in practice (for $n > 2$), on p. 17 of Dualities and adjunctions of stable derivators Beckert gives an example of a family of infinite chains of adjunctions that have period $4n + 2$ for every natural number $n$.