Are there non-trivial infinite chains of adjoint functors?

Question

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?

Answer 1

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:

We clarify the relationship between Grothendieck duality `a la Neeman and the Wirthm\"uller isomorphism `a la Fausk-Hu-May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: There exist either exactly three adjoints, exactly five, or infinitely many.

Answer 2

Broadening the question a bit, you can ask the same question about adjoint 1-morphisms in a 2-category (you're asking about 1-morphisms in the 2-category of categories). Then the 2-dimensional framed bordism 2-category gives a great example which is relatively elementary to understand. Furthermore it "explains" many of the other examples in a sense.

A bordism 2-category is one where the objects are 0-manifolds, the 1-morphisms are 1-manifolds with boundary, and the 2-morphisms are 2-manifolds with corners. Composition is gluing. This comes in several flavors depending on what kind of structure you put on the manifolds (oriented, spin, etc.). For technical reasons related to gluing structured manifolds, on the low-dimensional pieces you should put topological structure on a small collar $M^d \times \mathbb{R}^{2-d}$. A framing is a trivialization of the tangent bundle. Note that an interval has lots of framings in this stabilized sense, essentially one for each element of $\pi_1(\mathrm{SO}(2)) = \mathbb{Z}$.

One of the basic 1-morphisms is the interval with two incoming points (one positively framed and one negatively framed). Again this has a $\mathbb{Z}$-worth of framings. Taking adjoint switches it to having two outgoing points. But it's not hard to work out that the left and right adjoints have different framings!

We explain all of this in some detail in Section 2.2 of Dualizable Tensor Categories. (It's certainly not original to us, but that section has a lot more pictures and details than other places you can read about it.) The key image is:

Now, why is this relevant to Qiaochu's answer? A 2-dimensional local framed topological field theory is a symmetric monoidal functor from the 2-dimensional framed bordism category to another 2-category. In particular, we might hope that the image of interval would give a functor with a $\mathbb{Z}$ worth of framings, provided that changing framing acted nontrivially. Changing framing is composing with the interval with one incoming and one outgoing boundary and a single framing twist. It turns out that everything you would call as "Serre functor" appears in exactly this way. So Qiaochu's example is the image of this example under a TFT.

Answer 3

Another widely used example of infinite adjunction chains arises in linguistics, specifically in connection with pregroup grammars. It has been first observed by Lambek in Some Galois Connections in Elementary Number Theory (J. Number Theory 47 (1994), 371-377). Take monotone maps $f:\mathbb Z\to\mathbb Z$ unbounded in both directions (monotone with respect to the usual linear order on $\mathbb Z$). Every such map has both a left and a right adjoint and they rarely repeat, most such maps will produce infinite adjunction chains in both directions.

Answer 4

Another nice example is the infinite sequence of adjunctions characterizing stable homotopy theories. One has, in any homotopy theory $K$ (whatever you think that is, as long as there is a notion of (homotopy) limits and colimits) a sequence $1^*\vdash \pi^*\vdash 0^*$ of adjunctions between $K$ and the homotopy theory $K^{[1]}$ of morphisms in $K$, where's $1^*$ and $0^*$ evaluate an arrow at its codomain, respectively domain, and $\pi^*$ sends an object to its identity arrow. If $K$ has an initial and a final object, we can add an extra left adjoint $1_!$ and an extra right adjoint $0_*$ to the chain. If $K$ is finitely complete and cocomplete then it's pointed if and only if there are cone and fiber functors $C$ and $F$ extending the chain to length $7$.

Finally, $K$ is stable if and only if our chain extends infinitely in both directions: we use the loop and suspension and the isomorphism $C\cong \Sigma F$ to get the sequence of adjunctions $$...\vdash \Sigma 0^*\vdash 0_*\Omega \vdash C \vdash 1_!\vdash 1^*\vdash \pi^*\vdash 0^* \vdash 0_* \vdash F\vdash 1_!\Sigma \vdash \Omega 1^*..$$ While these are adjunctions of homotopy theories, they also give examples of adjunctions of categories by passing to the homotopy categories.

Proofs when $K$ is a derivator are in Section 4 of Groth's recent Characterization of abstract stable homotopy theories.

This sequence is still not far from cyclic; it comes from $C$ and $F$ factoring through each other up to functors $\Sigma$ and $\Omega$ with infinitely many left and right adjoints, respectively. But since $\Sigma$ and $\Omega$ sit in a cyclic adjunction sequence, it's still an essentially formal example.

Answer 5

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.