adjoint triple

This entry is about the notion of adjoint triple involving 3 functors. Please do not mix 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

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

is a triple of functors 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.



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


Every adjoint triple


gives rise to an adjoint pair

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

consisting of a monad GFG F left adjoint to the comonad GHG H on CC;

as well as an adjoint pair

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

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).)

Fully faithful adjoint triples


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


By a basic property of adjoint functors we have that

Moreover, by note 2 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 (for instance (Elephant, lemma 1.1.1) GHG 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 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 1, 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 2, 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.


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.


Some remarks on adjoint triples are in

On spherical triples see

Generalities are in

Proofs of the folklore Proposition 1 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.

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).

Revised on September 21, 2014 21:13:19 by Keith Harbaugh? (