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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.)

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  • $\begingroup$ I've wondered about this myself, and vaguely recall being told by a not entirely reliable source that there's some sort of "almost periodicity" that happens, though I wouldn't bet too heavily on it being true. Anyone with actual knowledge instead of half remembered hearsay? $\endgroup$ Dec 27, 2009 at 4:24
  • $\begingroup$ This much later question has basically interchangeable answers: mathoverflow.net/questions/242390/… $\endgroup$ Oct 21, 2018 at 5:25

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

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  • $\begingroup$ Thanks! Unfortunately, I won't be able to read this article until I get back to MIT. $\endgroup$ Dec 27, 2009 at 4:49
  • $\begingroup$ ssh linerva.mit.edu , w3m springerlink.com/content/pmj5074147116273 , scp file to your local machine... $\endgroup$ Dec 27, 2009 at 5:04
  • $\begingroup$ It's even easier to use EZproxy: go to libraries.mit.edu/about/faqs/… and add the bookmarklet to your browser toolbar now. (Harvard has it too but I don't know the URL offhand.) $\endgroup$ Dec 27, 2009 at 5:36
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    $\begingroup$ The link to springerlink.com is broken. Perhaps you could take a look, whenever possible... $\endgroup$ May 20, 2022 at 22:40
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    $\begingroup$ Downvoting because you didn't even include a title, which makes the paper impossible to identify now that the link is broken. $\endgroup$
    – varkor
    Jun 9, 2023 at 10:00
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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.

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  • $\begingroup$ I just noticed a couple weeks ago that the simplex category is actually a 2-category. Have you seen it used anywhere as such? $\endgroup$ Jan 1, 2010 at 3:18
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    $\begingroup$ Yes. You probably know that as a monoidal category, the simplex category is characterized by the universal property that it is initial among strict monoidal categories equipped with a monoid object. This is the key observation underlying the bar construction, among other things. The simplex 2-category has a similar 2-universal property, where we consider instead monoidal 2-categories equipped with a "KZ" monad, where the multiplication m: M@M --> M is left adjoint to the unit u@M: M --> M@M. Examples include cocompletion monads. Try googling this with Anders Kock (the K in KZ) as a key word. $\endgroup$
    – Todd Trimble
    Jan 1, 2010 at 3:59
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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 xi for distinct i, and so the functors xi ⊗ – (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.

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    $\begingroup$ 4-periodicity would be very interesting, since it is precisely the behavior that the Fourier transform enjoys. Hmm. $\endgroup$ Dec 27, 2009 at 5:15
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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,

  1. 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".

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