I am looking for examples of sequences of adjoint functors. That are (possibly bounded) sequences
$$(…,F_{-1}, F_{0}, F_1, F_2,…)$$
such that each $F_n$ is left adjoint to $F_{n+1}$. We call such a sequence cyclic of order $k$ if for one $n$ (and hence for all) we have $F_{n} \cong F_{n+k}$.
It is relatively easy to prove that cyclic sequences of all orders and non-cyclic sequences of all possible length exist. This can e.g. be done using posets, see http://www.springerlink.com/content/pmj5074147116273/.
I am looing for more "natural" examples of such sequences that are as long as possible. By natural I mean that they grow out of "usual functors" (sorry for this vague statement…)
Let my give two short examples:
1) Let $U: Top \to Set$ be the forgetful functor from locally connected topological spaces to sets. This induces a sequence of length 4:
$$ (\pi_0 , Dis , U , CoDis) $$
where $Dis$ and $CoDis$ are the functor that equip a set with the discrete and indiscrete topology. Then the sequence stops. Tons of examples of this type are induced by pullback functors in algebraic geometry.
2) a cyclic sequence of order 2: the Diagaonal functor $\Delta: A \to A \times A$ for any abelian category $A$ is left and right adjoint to the direct sum
$$ ( …,\Delta,\oplus,\Delta,\oplus,…)$$
Best Answer
The functor from the category of abelian groups to the category of arrows of abelian groups that sends an object to its identity morphism has three adjoints to the left and three to the right, for a chain of seven functors. The extreme adjoints are the functors that assign to an arrow its kernel or cokernel, as an object.