nLab adjoint string

Definition

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):

$$f_1 \dashv f_2 \dashv \cdots \dashv f_n$$

Special cases

• An adjoint $2$-tuple is just an ordinary adjunction.
• An adjoint $3$-tuple is an adjoint triple.
• An adjoint $4$-tuple is an adjoint quadruple.

Examples

1. 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.

2. 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.

3. 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.

4. 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).

5. 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.

6. Let $C$ be a category with a terminal object but no initial object. Then there are functors

   $$\array{ \delta_i \colon [n+1,C] \to [n,C] & 0\leq i \leq n; \\ \sigma_i\colon [n,C] \to [n+1,C] & 0\leq i \leq n }$$

   such that

   $$\delta_0 \dashv \sigma_0 \dashv \cdots \dashv \delta_n \dashv \sigma_n$$

   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 $C$).

7. 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

   $$P^{n-1} m \dashv P^{n-1} u P \dashv P^{n-2}m P \dashv \ldots \dashv m P^{n-1} \dashv u P^n$$

   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.

8. 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

   $$\ldots \dashv F \dashv G \dashv F \dashv \ldots$$

   of period 2.

References

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:

• Peter I. Booth, Sequences of adjoint functors, Archiv der Mathematik 23 (1972) 489-493 [doi:10.1007/BF01304920]

See also:

• Rudolf-E. Hoffmann, Sequences of adjoints for Ens-valued functors, manuscripta mathematica 32 (1980) [EuDML;154724]

Characterizing the category of sets as that whose Yoneda embedding extends to the left to an adjoint quintuple of adjoint functors:

• Robert Rosebrugh, Richard J. Wood, An adjoint characterization of the category of sets. PAMS 122 2 (1994) 409-413 [jstor:2161031]

On adjoint quadruples with a fully faithful right adjoint:

• Bob Rosebrugh, Richard J. Wood, Distributive Adjoint Strings, Theory and Applications of Categories, 1 6 (1995) 119-145 [tac:1-06]

See also:

• Moritz Groth, Mike Shulman, Generalized stability for abstract homotopy theories, arXiv:1704.08084.

On adjoint strings between a topos and Set and specifically on completely connected toposes (locally connected toposes with a further left adjoint to the left adjoint to the inverse image functor):

• Ryuya Hora: Grothendieck topos with a left adjoint to a left adjoint to a left adjoint to the global sections functor [arXiv:2503.04317]