ct.category theory - Iterated adjoint functors

A simple example where there are adjoint strings of arbitrary length is given by the simplex category, or rather the simplex 2-category, the sub-2-category of Cat whose objects are finite ordinals (so the 1-cells or functors are order-preserving maps, and the 2-cells or transformations are instances of the order relation f ≤ g). Notice that the functor 0: [1] --> [2] = {0, 1} is left adjoint to the unique functor !: [2] --> [1] which is left adjoint to the functor 1: [1] --> [2] = {0, 1}.

Using this and the monoidal structure, you can generate adjoint strings of arbitrary length which zig-zag between the cofaces i_k: [n] --> [n+1] and codegeneracies p_k: [n+1] --> [n]. Specifically, if i_0 < i_1 < ... < i_n name the n+1 injections [n] --> [n+1] and p_1 < ... < p_n name the n surjections [n+1] --> [n], then there is an adjoint string of the form

$i_0 dashv p_1 dashv i_1 dashv ldots dashv p_n dashv i_n$

and clearly there is no periodicity here.