In category theory, an adjoint string of length , adjoint chain of length , adjoint sequence of length , or an adjoint -tuple, is a sequence of adjunctions between functors (or more generally morphisms in a 2-category):
There is an adjoint -tuple between and . Indeed, given a locally small category , and the Yoneda embedding, , then being the rightmost functor of an adjoint -tuple entails that is equivalent to Set; see Rosebrugh-Wood.
For any category , there is a functor from 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 . If has an initial object , then has a further left adjoint assigning to each object the morphism ; and dually if has a terminal object then has a further right adjoint assigning to the morphism . Thus if has an initial and terminal object, we have an adjoint -tuple.
Continuing from the last example, if is moreover a pointed category with pullbacks and pushouts, then has a further left adjoint that constructs the cokernel of a morphism , i.e. the pushout of ; and has a further right adjoint that constructs the kernel of a morphism , namely the pullback of . Thus we have an adjoint -tuple. In fact, the existence of such an adjoint -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 -tuple to a -tuple again characterizes the pointed derivators. Moreover, the stable derivators are characterized by the extension of this -tuple to a doubly-infinite adjoint string with period 6 (GrothShul17).
Let denote the totally ordered -element set, regarded as a category. For each positive integer , we have order-preserving injections from to , and order-preserving surjections from to . Regarded as functors, these injections and surjections interleave to form an adjoint chain of length . These categories, functors, and adjunctions form the simplex category regarded as a locally posetal 2-category; see below.
Let 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 are obtained by applying to the simplex category example, and exploits the presence of the terminal object of ).
Generalizing the simplex category example: if is a lax idempotent monad with unit and multiplication (so that ), then there is an adjoint string
of length , back and forth between and . The example of and above is based on the fact that the simplex category , regarded as a locally posetal bicategory, is the walking lax idempotent monoid.
Given an ambidextrous adjunction (and in particular a self-adjoint functor), and , 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.