nLab adjoint triple


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)

(FGH):CD (F \dashv G \dashv H) \colon C \to D

is a triple of functors/morphisms F,H:CDF,H \colon C \to D and G:DCG \colon D \to C together with adjunction data FGF\dashv G and GHG\dashv H. That is, it is an adjoint string of length 3.


An adjoint triple (FGH)(F\dashv G\dashv H), def. is equivalently an adjoint pair in the 2-category whose morphisms are adjoint pairs in the original 2-category, hence an adjunction of adjunctions

(FG)(GH). (F \dashv G) \dashv (G \dashv H) \,.

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

F G G H \array{ F &\dashv& G \\ \bot && \bot \\ G &\dashv& H }

such that the two adjoint pairs appear horizontally, while the second order adjunction between them runs vertically.



The two adjunctions imply of course that GG preserves all limits and colimits that exist in DD.


Every adjoint triple

(FGH):CD (F \dashv G \dashv H) \colon C \to D

gives rise to an adjoint pair

(GFGH):CC (G F \dashv G H) \colon C \to C

consisting of the monad GFG F left adjoint to the comonad GHG H on CC; as well as to an adjoint pair

(FGHG):DD (F G \dashv H G) \colon D \to D

consisting of the comonad FGF G left adjoint to the monad HGH G on DD.

See adjoint monad for more.

In general there is a duality (an antiequivalence of categories) between the categories of monads having right adjoints and of comonads having left adjoints.

Note also (see there) that the algebras over a left-adjoint monad are identified with the coalgebras for its right-adjoint comonad (duel to Eilenberg & Moore 1965, see eg. MacLane & Moerdijk 1992, Theorems V.8.1 and V.8.2).

Fully faithful adjoint triples


For an adjoint triple FGHF\dashv G\dashv H we have that FF is fully faithful if and only if HH is fully faithful.

(In this case either adjoint pair is an idempotent adjunction, see Prop. below.)


By a basic property of adjoint functors (this Prop.), we have that

Moreover, by Note and the fact that adjoints are unique up to isomorphism, we have that GFG F is isomorphic to the identity precisely if GHG H is.

Finally, by a standard fact about adjoint functors (see for instance (Elephant, Lemma A1.1.1)), we have that GHG H is naturally isomorphic to the identity precisely if it is so by the counit.

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 GHIdG H \to Id or the unit IdGFId \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 , we say that FGHF\dashv G \dashv H is a fully faithful adjoint triple. This is often the case when DD is a category of “spaces” structured over CC, where FF and HH construct “discrete” and “codiscrete” spaces respectively.

For instance, if G:DCG\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, GG need not be faithful), but they do exhibit certain similar phenomena. In particular, we have the following.


Suppose (FGH):CD(F \dashv G \dashv H) \colon C \to D is an adjoint triple in which FF and HH are fully faithful, and suppose that CC is cocomplete. Then GG admits final lifts for small GG-structured sinks.


Let {G(S i)X}\{G(S_i) \to X\} be a small sink in CC, and consider the diagram in DD consisting of all the S iS_i, all the counits ε:FG(S i)S i\varepsilon\colon F G(S_i) \to S_i (where FF is the left adjoint of GG), and all the images FG(S i)F(X)F G(S_i) \to F(X) of the morphisms making up the sink. The colimit of this diagram is preserved by GG (since it has a right adjoint as well). But the image of the diagram consists essentially of just the sink itself (since FF is fully faithful, G(ε)G(\varepsilon) is an isomorphism), and its colimit is XX; hence the colimit of the original diagram is a lifting of XX to DD (up to isomorphism). It is easy to verify that this lifting has the correct universal property.

Thus, we can talk about objects of DD having the weak structure or strong structure induced by any small collection of maps.


In the situation of Proposition , GG 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 CC is complete, then GG admits initial lifts for small GG-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.

Idempotent adjoint triples


For an adjoint triple FGHF\dashv G\dashv H, the adjunction FGF\dashv G is an idempotent adjunction if and only if the adjunction GHG\dashv H is so.


The monad GFG F is left adjoint to the comonad GHG H, with the structure maps being mates. Therefore, by a standard fact, the category of GFG F-algebras and the category of GHG H-coalgebras are isomorphic over their common base. However, FGF\dashv G is idempotent precisely when GFG F is an idempotent monad, hence precisely when the forgetful functor of the category of GFG F-algebras is fully faithful, and dually for GHG\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.

A special case of this situation is Prop. above.


Special cases

Specific examples

  • Given any ring homomorphism f :RSf^\circ: R\to S (in commutative case dual to an affine morphism f:SpecSSpecRf: Spec S\to Spec R of affine schemes), there is an adjoint triple f !f *f *f^!\dashv f_*\dashv f^* where f *: RMod SModf^*: {}_R Mod\to {}_S Mod is an extension of scalars, f *: SMod RModf_*: {}_S Mod\to {}_R Mod the restriction of scalars and f !:MHom R( RS, RM)f^! : M\mapsto Hom_R ({}_R S, {}_R M) its right adjoint. This triple is affine in the above sense.

  • If TT is a lax-idempotent 2-monad, then a TT-algebra AA has an adjunction a:TAA:η Aa : T A \rightleftarrows A : \eta_A. If this extends to an adjoint triple with a further left adjoint to aa, then AA is called a continuous algebra.

  • See essential geometric morphism.


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 can be found in

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).

Last revised on April 1, 2024 at 07:40:57. See the history of this page for a list of all contributions to it.