TITLE: Are there non-trivial infinite chains of adjoint functors?
QUESTION [26 upvotes]: There are self-adjoint functors $A \dashv A$. There are also functors $A$ that are both left- and right-adjoint to another functor $B$. $$A \dashv B \dashv A$$
There are also finite cyclic chains of adjoint functors, that can have the initial functor equal to the final functor in the chain. $$A_0 \dashv A_1 \dashv \dotsb \dashv A_{n-1} \dashv A_0$$Not including those cases, are there nontrivial examples of infinite chains of adjoint functors, going infinitely to the left and infinitely to the right? $$ \dotsb \dashv A_{n-1} \dashv A_n \dashv A_{n+1} \dashv \dotsb $$ 
Does it make sense to consider such a chain as being an analogue of a long exact sequence, and if so, does it make sense to consider the monads $A_{n+1} \circ A_n$ and comonads $A_{n-1} \circ A_n$formed by composition of adjacent pairs to be at all analogous to homology or cohomology groups generated by long exact sequences? Is this a useful concept? Do these infinite chains appear often enough to warrant such an analogy?

REPLY [4 votes]: My paper Non-symmetric $*$-autonomous categories.  Theoretical
Computer Science, {\bf139} (1995), 115-130, might be thought of as a generalization of Lambek's to Chu categories.  It certainly leads to doubly infinite sequences of adjunctions.