nLab adjoint string

Adjoint strings

Adjoint strings

Definition

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

Examples

  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.

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:

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.