An adjoint quadruple is a sequence of three adjunctions
between a quadruple of morphisms.
Every adjoint quadruple
induces an adjoint triple on
(hence a monad left adjoint to a comonad left adjoint to a monad) and an adjoint triple
Since moreover every adjoint triple induces an adjoint pair and an adjoint pair , the adjoint quadruple above induces four adjoint pairs, such as
Canonical natural transformations
Let be an adjoint quadruple of adjoint functors such that and are full and faithful functors. We record some general properties of such a setup.
etc. for units and
etc. for counits.
We have commuting diagrams, natural in ,
where the diagonal morphisms
are defined to be the equal composites of the sides of these diagrams.
This appears as (Johnstone, lemma 2.1, corollary 2.2).
The following conditions are equivalent:
for all the morphism is an epimorphism;
for all , the morphism is a monomorphism;
is faithful on morphisms of the form .
This appears as (Johnstone, lemma 2.3).
By the above definition, is a monomorphism precisely if is. This in turn is so (see monomorphism) precisely if the first function in
and hence the composite is a monomorphism in Set.
By definition of adjunct and using the -zig-zag identity, this is equal to the action of on morphisms
Similarly, by the above definition the morphism is an epimorphism precisely if is so, which is the case precisely if the top morphism in
and hence the bottom morphism is a monomorphism in Set, where again the commutativity of this diagram follows from the definition of adjunct and the -zig-zag identity.
For any pair of adjoint functors, there is induced on the presheaf categories a quadruple of adjoint functors
where and denote left and right Kan extension, respectively.
For cohesive topos by definition the terminal geometric morphism extends to an adjoint quadruple.