nLab adjoint string

Adjoint strings

Adjoint strings


In category theory, an adjoint string of length nn, adjoint chain of length nn, adjoint sequence of length nn, or an adjoint nn-tuple, is a sequence of (n1)(n-1) adjunctions between nn functors (or more generally morphisms in a 2-category):

f 1f 2f nf_1 \dashv f_2 \dashv \cdots \dashv f_n

Special cases


  1. There is an adjoint 55-tuple between [Set op,Set][Set^{op}, Set] and SetSet. Indeed, given a locally small category BB, and the Yoneda embedding, y:B[B op,Set]y: B \to [B^{op}, Set], then yy being the rightmost functor of an adjoint 55-tuple entails that BB is equivalent to Set; see Rosebrugh-Wood.

  2. For any category CC, there is a functor ids:CAr(C)ids: C\to Ar(C) from CC 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 codidsdomcod \dashv ids \dashv dom. If CC has an initial object 00, then codcod has a further left adjoint II assigning to each object xx the morphism 0x0\to x; and dually if CC has a terminal object 11 then domdom has a further right adjoint TT assigning to xx the morphism x1x\to 1. Thus if CC has an initial and terminal object, we have an adjoint 55-tuple.

  3. Continuing from the last example, if CC is moreover a pointed category with pullbacks and pushouts, then II has a further left adjoint that constructs the cokernel of a morphism xyx\to y, i.e. the pushout of yx0y \leftarrow x \to 0; and TT has a further right adjoint that constructs the kernel of a morphism xyx \to y, namely the pullback of xy0x\to y \leftarrow 0. Thus we have an adjoint 77-tuple. In fact, the existence of such an adjoint 77-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 55-tuple to a 77-tuple again characterizes the pointed derivators. Moreover, the stable derivators are characterized by the extension of this 77-tuple to a doubly-infinite adjoint string with period 6 (GrothShul17).

  5. Let [n][n] denote the totally ordered (n+1)(n+1)-element set, regarded as a category. For each positive integer nn, we have n+1n+1 order-preserving injections from [n1][n-1] to [n][n], and nn order-preserving surjections from [n][n] to [n1][n-1]. Regarded as functors, these injections and surjections interleave to form an adjoint chain of length 2n+12n + 1. These categories, functors, and adjunctions form the simplex category regarded as a locally posetal 2-category; see below.

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

    δ i:[n+1,C][n,C] 0in; σ i:[n,C][n+1,C] 0in \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

    δ 0σ 0δ nσ n \delta_0 \dashv \sigma_0 \dashv \cdots \dashv \delta_n \dashv \sigma_n

    is a maximal string of adjoint functors (all but σ n\sigma_n are obtained by applying [,C][-, C] to the simplex category example, and σ n\sigma_n exploits the presence of the terminal object of CC).

  7. Generalizing the simplex category example: if PP is a lax idempotent monad with unit u:1Pu: 1 \to P and multiplication m:PPPm: P P \to P (so that muPm \dashv u P), then there is an adjoint string

    P n1mP n1uPP n2mPmP n1uP nP^{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 2n+12 n + 1, back and forth between P n+1P^{n+1} and P nP^n. The example of [n][n] and [n+1][n+1] above is based on the fact that the simplex category Δ\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), FGF \dashv G and GFG \dashv F, we of course get an infinite adjoint string

    FGF\ldots \dashv F \dashv G \dashv F \dashv \ldots

    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.