Iterated adjoint functors

Question

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.)

Answer 1

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

Answer 2

A simple example where there are adjoint strings of arbitrary length is given by the simplex category, or rather the simplex 2-category, the sub-2-category of Cat whose objects are finite ordinals (so the 1-cells or functors are order-preserving maps, and the 2-cells 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 zig-zag 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.

Answer 3

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 x₀, 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 4-periodic. One that I think is true is when you are in a 3-category and all your unit and counit 2-morphisms 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 3-category with adjoints.

Answer 4

Let $C$ be an $\infty$-category.

1. If $C$ has a zero object, then the unique functor $C \to pt$ is an ambidextrous adjoint (i.e. fits into an adjoint string periodic of order 2).

2. (If $C$ is semiadditive, then the diagonal functor $C \to C^2$ likewise is an ambidextrous adjunction.)

3. Let $C^{[1]}$ be the category of arrows of $C$. There is always an adjunction $cod \dashv id \dashv dom$ between the codomain functor, the functor which takes the identity arrow, and the domain functor. If $C$ has an initial object, this chain extends one step further to the left, and if $C$ has a terminal object, it extends one further to the right. If $C$ is pointed and has cokernels it extends an additional step to the left, and if $C$ has kernels it extends an additional step to the right.

   If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 6 up to a twist by the suspension functor $\Sigma$.

4. Likewise, let $C^{[2]}$ be the category of composable pairs of arrows of $C$. The composition map $C^{[2]} \to C^{[1]}$ always fits into the middle of an adjoint string of length 5, which extends an additional step with intitial / terminal objects and again with kernels / cokernels.

   If $C$ is stable, this adjoint string extends infinitely in both directions. It is periodic of order 24 up to a twist by $\Sigma^3$.

Examples $(1,3,4)$ clearly suggest that if $C$ is stable, then between $C^{[n-1]}$ and $C^{[n]}$ we have a bi-infinite adjoint string, perhaps periodic (up to a shift) of order $(n+1)!$. But I'm not sure.

Various reindexing functors for functor categories into a stable $\infty$-category from various other finite categories also induce infinite adjoint strings. For instance,

5. the constant functor $C \to C^{\bullet \leftarrow \bullet \to \bullet}$ fits in a bi-infinite adjoint string which is periodic of order 4 up to a shift by $\Sigma$.

If $C$ is ambidextrous in the sense of (Hopkins and) Lurie, even more functor categories do this.

Another place this happens is in Grothendieck duality -- it turns out that an exact symmetric monoidal functor between nice tensor-triangulated categories either fits in an adjoint string of length 1, 3, 5, or bi-infinity. In the infinite case, it is periodic of order 6 up to a twist by the so-called "dualizing object".