TITLE: Examples of five-adjoint systems
QUESTION [9 upvotes]: I'm looking into Lawvere's formulation of unities of opposites and opposites of unities, and for this I would be interested in systems of five (or more) adjoint functors $X\stackrel{\stackrel{\longrightarrow}{\longleftarrow}}{\stackrel{\longrightarrow}{\stackrel{\longleftarrow}{\longrightarrow}}}Y$, ideally within the domains of topos theory and algebraic geometry, but really any examples would be interesting. I know they are very rare, but I feel like I have come across one at least once. Anyone?

REPLY [9 votes]: As noted by Simon Henry,  the nLab gives many examples of adjoint chains.  You can get an infinitely long adjoint chains from any ambidextrous adjunction.  For an example, let $G$ be a finite group and $H$ a subgroup of $G$.  Let $\mathrm{Rep}(G)$ be the category of finite-dimensional representations of $G$ over your favorite field, and similarly for $\mathrm{Rep}(H)$.  Then restriction gives a functor
$$ \mathrm{restriction} : \mathrm{Rep}(G) \to \mathrm{Rep}(H) $$
and let
$$ \mathrm{induction} : \mathrm{Rep}(H) \to \mathrm{Rep}(G) $$
be its left adjoint.   Induction is also the right adjoint of restriction, so we get an infinitely long adjoint chain
$$ \cdots \dashv \mathrm{induction} \dashv \mathrm{restriction} \dashv \mathrm{induction} \dashv \cdots $$
For this reason it might be more exciting to get adjoint strings of length 5, or more generally length $n$, that can't be extended further.  The nLab also gives plenty of those.
Amazingly, Rosebrugh and Wood showed that any category with an adjoint string of length 5
$$    U \dashv V \dashv W \dashv X \dashv Y $$
where $Y$ is its Yoneda embedding must be equivalent to the category of sets.   (More precisely, the category of small sets, where we assume the axiom of universes.)

Robert Rosebrugh and R. J. Wood, An adjoint characterization of the category of sets,  Proceedings of the American Mathematical Society 122, no. 2 (1994): 409–413.

REPLY [7 votes]: You can build long chains of adjoints by taking functor categories and using Kan extensions.  I will give an example.
Write $\underline{n}$ for the set $\{1, \ldots, n\}$ considered as a discrete category.  As $\underline{1}$ is the terminal category, there is a unique functor $f \colon \underline{2} \to \underline{1}$.
Precomposing with $f$ gives a functor $f^* \colon [\underline{1}, \mathrm{Set}] \to [\underline{2}, \mathrm{Set}]$.  As a functor $\underline{n} \to \mathrm{Set}$ is just an $n$-tuple of sets, we have that $f^*(X) = (X, X)$ so $f^*$ is the diagonal functor.  The functor $f^*$ has adjoints on both sides, which are the Kan extensions.  The left Kan $f_!$ provides a method of combining two sets $f_!(X,Y)$ so that a map $f_!(X,Y) \to Z$ is the same data as a pair of maps $(X \to Z, Y \to Z)$.  In other words, $f_!(X,Y) = X \sqcup Y$ is the disjoint union.  Similarly, the right Kan $f_*(Y,Z)$ provides a method of combining $Y$ and $Z$ so that $X \to f_*(Y,Z)$ is the same data as a pair of maps $(X \to Y, X \to Z)$.  In other words, $f_*(Y,Z) = Y \times Z$.  So far, we have a two-step adjunction of known useful functors
$$
f_! \dashv f^* \dashv f_*
$$
Precomposing with $f^*$ gives a functor $(f^*)^* \colon [[\underline{2},\mathrm{Set}], \mathrm{Set}] \to [[\underline{1},\mathrm{Set}], \mathrm{Set}]$.  As a functor $[\underline{2},\mathrm{Set}]\to \mathrm{Set}$ is an operation $\odot \colon \mathrm{Set} \times \mathrm{Set} \to \mathrm{Set}$ that takes two sets and returns a set, we have that $(f^*)^*(\odot)$ is the "squaring" functor $(X) \mapsto X \odot X$.
Certainly we have Kan extensions
$$
(f^*)_! \dashv (f^*)^* \dashv (f^*)_*
$$
However, if $L \dashv R$ is an adjunction, we always have $L_! \cong R^*$ and $L^* \cong R_*$.  Consequently, we can build the five-functor chain
$$
(f_*)_! \dashv (f_*)^* = (f^*)_! \dashv (f^*)^* \dashv (f^*)_* = (f_!)^* \dashv (f_!)_*
$$
For example, the functor $(f^*)_!$ takes a unary operation $\Gamma \colon \mathrm{Set} \to \mathrm{Set}$, considers it as a functor defined on the diagonal $\mathrm{Set} \overset{\Delta}{\to} \mathrm{Set} \times \mathrm{Set}$, and freely generates from here a binary operation.  By the formula $L_! \dashv R^*$, we can calculate this rather concretely as sending $\Gamma$ to the binary operation $X \otimes_\Gamma Y = \Gamma(X \times Y)$.  A natural transformation of bifunctors $\otimes_\Gamma \implies \odot$ will somehow be the same information as a natural transformation $\Gamma \implies (X \mapsto X \odot X)$.
We could evidently continue this pattern to build even longer and less-understandable chains.