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!
An adjoint triple (of functors between categories or generally of 1-morphisms in a 2-category)
is a triple of functors/morphisms $F,H \colon C \to D$ and $G \colon D \to C$ together with adjunction data $F\dashv G$ and $G\dashv H$. That is, it is an adjoint string of length 3.
An adjoint triple $(F\dashv G\dashv H)$, def. 1 is equivalently an adjoint pair in the 2-category whose morphisms are adjoint pairs in the original 2-category, hence an adjunction of adjunctions
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.
The two adjunctions imply of course that $G$ preserves all limits and colimits that exist in $D$.
Every adjoint triple
gives rise to an adjoint pair
consisting of a monad $G F$ left adjoint to the comonad $G H$ on $C$;
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 $F\dashv G\dashv H$ we have that $F$ is fully faithful precisely if $H$ is fully faithful.
By a basic property of adjoint functors we have that
the left adjoint $F$ being full and faithful is equivalent to the unit $Id \to G F$ being a natural isomorphism;
the right adjoint $H$ being full and faithful is equivalent to the counit $G H \to Id$ being a natural isomorphism.
Moreover, by note 2 and the fact that adjoints are unique up to isomorphism, we have that $G F$ is isomorphic to the identity precisely if $G H$ is.
Finally, by a standard fact about adjoint functors (for instance (Elephant, lemma 1.1.1) $G H$ is isomorphic to the identity precisely if it is so by the adjunction unit.
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 $G H \to Id$ or the unit $Id \to G F$ 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 $F\dashv G \dashv H$ is a fully faithful adjoint triple. This is often the case when $D$ is a category of “spaces” structured over $C$, where $F$ and $H$ construct “discrete” and “codiscrete” spaces respectively.
For instance, if $G\colon D\to C$ 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, $G$ need not be faithful), but they do exhibit certain similar phenomena. In particular, we have the following.
Suppose $(F \dashv G \dashv H) \colon C \to D$ is an adjoint triple in which $F$ and $H$ are fully faithful, and suppose that $C$ is cocomplete. Then $G$ admits final lifts for small $G$-structured sinks.
Let $\{G(S_i) \to X\}$ be a small sink in $C$, and consider the diagram in $D$ consisting of all the $S_i$, all the counits $\varepsilon\colon F G(S_i) \to S_i$ (where $F$ is the left adjoint of $G$), and all the images $F G(S_i) \to F(X)$ of the morphisms making up the sink. The colimit of this diagram is preserved by $G$ (since it has a right adjoint as well). But the image of the diagram consists essentially of just the sink itself (since $F$ is fully faithful, $G(\varepsilon)$ is an isomorphism), and its colimit is $X$; hence the colimit of the original diagram is a lifting of $X$ to $D$ (up to isomorphism). It is easy to verify that this lifting has the correct universal property.
Thus, we can talk about objects of $D$ having the weak structure or strong structure induced by any small collection of maps.
In the situation of Proposition 3, $G$ is a (Street) opfibration. If it is also an isofibration, then it is a Grothendieck opfibration.
A final lift of a singleton sink is precisely an opcartesian arrow.
Dually, of course, if $C$ is complete, then $G$ admits initial lifts for small $G$-structured cosinks and is a fibration.
In particular, the proposition and its corollary apply to a cohesive topos, and (suitably categorified) to a cohesive (∞,1)-topos.
For an adjoint triple $F\dashv G\dashv H$, the adjunction $F\dashv G$ is an idempotent adjunction if and only if the adjunction $G\dashv H$ is so.
The monad $G F$ is left adjoint to the comonad $H G$, with the structure maps being mates. Therefore, by a standard fact, the category of $G F$-algebras and the category of $H G$-coalgebras are isomorphic over their common base. However, $F\dashv G$ is idempotent precisely when $G F$ is an idempotent monad, hence precisely when the forgetful functor of the category of $G F$-algebras is fully faithful, and dually for $G\dashv H$. Since the categories of algebras are isomorphic respecting their forgetful functors, one forgetful functor is fully faithful if and only if the other is.
If one of the two adjoint pairs induced from an adjoint triple involving identities, then the other exhibits an adjoint cylinder / unity of opposites.
An adjoint triple $F\dashv G\dashv H$ is Frobenius if $F$ is naturally isomorphic to $H$. 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 $A_\infty$- 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.
A context of six operations $(f_! \dashv f^!)$, $(f^\ast \dashv f_\ast)$ induces an adjoint triple when either $f^! \simeq f^\ast$ or $f_! = f_\ast$. This is called a Wirthmüller context or a Grothendieck context, respectively.
Given any ring homomorphism $f^\circ: R\to S$ (in commutative case dual to an affine morphism $f: Spec S\to Spec R$ of affine schemes), there is an adjoint triple $f^!\dashv f_*\dashv f^*$ where $f^*: {}_R Mod\to {}_S Mod$ is an extension of scalars, $f_*: {}_S Mod\to {}_R Mod$ the restriction of scalars and $f^! : M\mapsto Hom_R ({}_R S, {}_R M)$ its right adjoint. This triple is affine in the above sense.
If $T$ is a lax-idempotent 2-monad, then a $T$-algebra $A$ has an adjunction $a : T A \rightleftarrows A : \eta_A$. If this extends to an adjoint triple with a further left adjoint to $a$, then $A$ is called a continuous algebra.
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.
G.M. Kelly and F.W. Lawvere, “On the complete lattice of essential localizations”, Bulletin de la Société Mathématique de Belgique, Série A, v. 41 no 2 (1989) 289–319.
Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic, (1992) (Lemma 7.4.1).
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).