This entry is about the notion of adjoint triple involving three functors. This is not to be confused with the notion of adjoint monads, which were also sometimes called adjoint triples, with “triple” then being a synonym for “monad”. However, an adjoint triple in the sense here does induce an adjoint monad!
This fact plays an important role in Licata-Shulman, 5.1. Relatedly, it also appears in the characterization of certain kinds of geometric morphism (e.g. the local ones) in terms of adjunctions in the 2-category Topos.
It may be suggestive to denote this like so
such that the two adjoint pairs appear horizontally, while the second order adjunction between them runs vertically.
Every adjoint triple
gives rise to an adjoint pair
as well as an adjoint pair
See adjoint monad for more.
In general there is a duality (an antiequivalence of categories) between the category of monads having right adjoints and comonads having left adjoints. Note also that the algebras for a left-adjoint monad can be identified with the coalgebras for its right adjoint comonad. (Theorems 5.8.1 and 5.8.2 in (SGL).)
For an adjoint triple we have that is fully faithful precisely if is fully faithful.
Moreover, by note 2 and the fact that adjoints are unique up to isomorphism, we have that is isomorphic to the identity precisely if is.
The preceeding proposition is folklore; perhaps its earliest appearance in print is (DT, Lemma 1.3). A slightly shorter proof is in (KL, Prop. 2.3). Both proofs explicitly exhibit an inverse to the counit or the unit given an inverse to the other (which could be extracted by beta-reducing the above, slightly more abstract argument). It also appears in (SGL, Lemma 7.4.1).
In the situation of Proposition 2, we say that is a fully faithful adjoint triple. This is often the case when is a category of “spaces” structured over , where and construct “discrete” and “codiscrete” spaces respectively.
For instance, if is a topological concrete category, then it has both a left and right adjoint which are fully faithful. Not every fully faithful adjoint triple is a topological concrete category (among other things, need not be faithful), but they do exhibit certain similar phenomena. In particular, we have the following.
Let be a small sink in , and consider the diagram in consisting of all the , all the counits (where is the left adjoint of ), and all the images of the morphisms making up the sink. The colimit of this diagram is preserved by (since it has a right adjoint as well). But the image of the diagram consists essentially of just the sink itself (since is fully faithful, is an isomorphism), and its colimit is ; hence the colimit of the original diagram is a lifting of to (up to isomorphism). It is easy to verify that this lifting has the correct universal property.
A final lift of a singleton sink is precisely an opcartesian arrow.
Dually, of course, if is complete, then admits initial lifts for small -structured cosinks and is a fibration.
For an adjoint triple , the adjunction is an idempotent adjunction if and only if the adjunction is so.
The monad is left adjoint to the comonad , with the structure maps being mates. Therefore, by a standard fact, the category of -algebras and the category of -coalgebras are isomorphic over their common base. However, is idempotent precisely when is an idempotent monad, hence precisely when the forgetful functor of the category of -algebras is fully faithful, and dually for . Since the categories of algebras are isomorphic respecting their forgetful functors, one forgetful functor is fully faithful if and only if the other is.
An adjoint triple is Frobenius if is naturally isomorphic to . See Frobenius functor.
An affine morphism is an adjoint triple of functors in which the middle term is conservative. For example, any affine morphism of schemes induce an affine triples of functors among the categories of quasicoherent modules.
An adjoint triple of functors among - or triangulated functors with certain additional structure is called spherical . See e.g. (Anno). The main examples come from Serre functors in a Calabi-Yau category context.
Given any ring homomorphism (in commutative case dual to an affine morphism of affine schemes), there is an adjoint triple where is an extension of scalars, the restriction of scalars and its right adjoint. This triple is affine in the above sense.
Some remarks on adjoint triples are in
The modal type theory of adjoint triples is discussed in
On spherical triples see
Generalities are in
Proofs of the folklore Proposition 2 can be found in
Roy Dyckhoff and Walter Tholen, “Exponentiable morphisms, partial products, and pullback complements”, JPAA 49 (1987), 103–116.
Several lemmas concerning adjoint pairs and adjoint triples are included in
together with geometric consequences. Note a somewhat nonstandard usage of terminology continuous functor (also flatness in the paper includes having right adjoint).