In category theory, an adjoint string of length $n$, adjoint chain of length $n$, adjoint sequence of length $n$, or an adjoint $n$-tuple, is a sequence of $(n-1)$ adjunctions between $n$ functors (or more generally morphisms in a 2-category):
There is an adjoint $5$-tuple between $[Set^{op}, Set]$ and $Set$. Indeed, given a locally small category $B$, and the Yoneda embedding, $y: B \to [B^{op}, Set]$, then $y$ being the rightmost functor of an adjoint $5$-tuple entails that $B$ is equivalent to Set; see Rosebrugh-Wood.
For any category $C$, there is a functor $ids: C\to Ar(C)$ from $C$ to its arrow category that assigns the identity morphism of each object. This functor always has both a left and a right adjoint which assign the codomain and domain of an arrow respectively; thus we have an adjoint triple $cod \dashv ids \dashv dom$. If $C$ has an initial object $0$, then $cod$ has a further left adjoint $I$ assigning to each object $x$ the morphism $0\to x$; and dually if $C$ has a terminal object $1$ then $dom$ has a further right adjoint $T$ assigning to $x$ the morphism $x\to 1$. Thus if $C$ has an initial and terminal object, we have an adjoint $5$-tuple.
Continuing from the last example, if $C$ is moreover a pointed category with pullbacks and pushouts, then $I$ has a further left adjoint that constructs the cokernel of a morphism $x\to y$, i.e. the pushout of $y \leftarrow x \to 0$; and $T$ has a further right adjoint that constructs the kernel of a morphism $x \to y$, namely the pullback of $x\to y \leftarrow 0$. Thus we have an adjoint $7$-tuple. In fact, the existence of such an adjoint $7$-tuple characterizes pointed categories among categories with finite limits and colimits.
The previous two examples apply also to derivators, and the extension of the analogous adjoint $5$-tuple to a $7$-tuple again characterizes the pointed derivators. Moreover, the stable derivators are characterized by the extension of this $7$-tuple to a doubly-infinite adjoint string with period 6 (GrothShul17).
Let $[n]$ denote the totally ordered $(n+1)$-element set, regarded as a category. For each positive integer $n$, we have $n+1$ order-preserving injections from $[n-1]$ to $[n]$, and $n$ order-preserving surjections from $[n]$ to $[n-1]$. Regarded as functors, these injections and surjections interleave to form an adjoint chain of length $2n + 1$. These categories, functors, and adjunctions form the simplex category regarded as a locally posetal 2-category; see below.
Let $C$ be a category with a terminal object but no initial object. Then there are functors
such that
is a maximal string of adjoint functors (all but $\sigma_n$ are obtained by applying $[-, C]$ to the simplex category example, and $\sigma_n$ exploits the presence of the terminal object of $C$).
Generalizing the simplex category example: if $P$ is a lax idempotent monad with unit $u: 1 \to P$ and multiplication $m: P P \to P$ (so that $m \dashv u P$), then there is an adjoint string
of length $2 n + 1$, back and forth between $P^{n+1}$ and $P^n$. The example of $[n]$ and $[n+1]$ above is based on the fact that the simplex category $\Delta$, regarded as a locally posetal bicategory, is the walking lax idempotent monoid.
Given an ambidextrous adjunction (and in particular a self-adjoint functor), $F \dashv G$ and $G \dashv F$, we of course get an infinite adjoint string
of period 2.
A study of adjoint strings, in particular showing that cyclic chains of any length, and adjoint chains of any length exist, may be found in:
See also:
Characterizing the category of sets as that whose Yoneda embedding extends to the left to an adjoint quintuple of adjoint functors:
On adjoint quadruples with a fully faithful right adjoint:
See also:
Last revised on November 16, 2023 at 10:44:58. See the history of this page for a list of all contributions to it.