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?
Best Answer
Let $C$ be a category enriched over finite-dimensional $k$-vector spaces. A Serre functor for $C$ is a $k$-linear automorphism $S : C \to C$ such that there is a natural equivalence
$$\text{Hom}(x, y) \cong \text{Hom}(y, Sx)^{\ast}.$$
Serre functors are unique when they exist. The example that motivates the name occurs when $C = D_b(X)$ is the bounded derived category of coherent sheaves on a smooth projective variety $X$ over $k$ of dimension $n$; in this case, the claim that $S(-) = (-) \otimes \omega_X[n]$ is a Serre functor on $C$ is Serre duality.
Let $C, D$ be categories which admit Serre functors $S_C, S_D$, and let $F : C \to D, G : D \to C$ be an adjunction between them, with $F$ the left adjoint and $G$ the right adjoint. Then we have
$$\text{Hom}_D(x, Fy) \cong \text{Hom}_D(Fy, S_D x)^{\ast} \cong \text{Hom}_C(y, GS_D x)^{\ast} \cong \text{Hom}_C(GS_D x, S_C y) \cong \text{Hom}_C(S_C^{-1} G S_D x, y)$$
from which it follows that $S_C^{-1} G S_D$ is the left adjoint of $F$. More generally, by iterating Serre functors we get an infinite (in both directions) chain of adjoints which are generally different, although as Dylan Wilson says they just differ by a "twist" (e.g. for smooth projective varieties, they differ by tensoring by an invertible object, namely a shift of the relative canonical bundle). This implies, in particular, that we don't get any new monads or comonads by continuing the chain.
Edit: Grothendieck-Neeman duality and the Wirthmüller isomorphism by Balmer, Dell'Ambrogio, and Sanders might be of interest. I think this is the paper Dylan refers to in the comments. Quoting from the abstract: