Let $F_0 : C \to D$ be a functor. If it exists, let $G_0 : D \to C$ be its left adjoint. If it exists, let $F_1 : C \to D$ be its left adjoint. And so forth. In situations where the infinite sequence $(F_0, G_0, F_1, G_1, ...)$ exists, when is it periodic? Aperiodic? (Feel free to replace all "lefts" by "rights," of course.)

http://www.springerlink.com/content/pmj5074147116273/ considers sequences of adjoint functors just like you describe. 


In general, all the functors might be nonisomorphic. The way I know how to prove this is to consider the free monoidal (not symmetric) category with left and right duals on a single object x_{0}, and show that there are no maps between the x_{i} for distinct i, and so the functors x_{i} ⊗ – (which form such a chain) are definitely distinct. I believe there are some natural situations however where the sequence is 4periodic. One that I think is true is when you are in a 3category and all your unit and counit 2morphisms themselves have adjoints. This must be true and the root reason is that taking the double left adjoint corresponds to the generator of $\pi_1(O(2)) = \mathbb{Z}$ but twice that generator is killed in $\pi_1(O(3)) = \mathbb{Z}/2$. But so far I haven't managed to turn this into a direct proof using the axioms of a 3category with adjoints. 


A simple example where there are adjoint strings of arbitrary length is given by the simplex category, or rather the simplex 2category, the sub2category of Cat whose objects are finite ordinals (so the 1cells or functors are orderpreserving maps, and the 2cells or transformations are instances of the order relation f ≤ g). Notice that the functor 0: [1] > [2] = {0, 1} is left adjoint to the unique functor !: [2] > [1] which is left adjoint to the functor 1: [1] > [2] = {0, 1}. Using this and the monoidal structure, you can generate adjoint strings of arbitrary length which zigzag between the cofaces i_k: [n] > [n+1] and codegeneracies p_k: [n+1] > [n]. Specifically, if i_0 < i_1 < ... < i_n name the n+1 injections [n] > [n+1] and p_1 < ... < p_n name the n surjections [n+1] > [n], then there is an adjoint string of the form $i_0 \dashv p_1 \dashv i_1 \dashv \ldots \dashv p_n \dashv i_n$ and clearly there is no periodicity here. 

