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\section*{adjoint triple}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\begin{quote}%
This entry is about the notion of \emph{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]]!


\end{quote}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{FullyFaithFulAdjointTriples}{Fully faithful adjoint triples}\dotfill \pageref*{FullyFaithFulAdjointTriples} \linebreak
\noindent\hyperlink{idempotent_adjoint_triples}{Idempotent adjoint triples}\dotfill \pageref*{idempotent_adjoint_triples} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{specific_examples}{Specific examples}\dotfill \pageref*{specific_examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{Defn}\hypertarget{Defn}{}
An \textbf{adjoint triple} (of [[functors]] between [[categories]] or generally of [[1-morphisms]] in a [[2-category]])

\begin{displaymath}
( F\dashv G\dashv H) \colon C \longrightarrow D
\end{displaymath}
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.

\end{defn}
\begin{prop}
\label{AsAdjunctionOfAdjunctions}\hypertarget{AsAdjunctionOfAdjunctions}{}
An adjoint triple $(F\dashv G\dashv H)$, def. \ref{Defn} is equivalently an [[adjoint pair]] in the 2-category whose morphisms are adjoint pairs in the original 2-category, hence an adjunction of adjunctions

\begin{displaymath}
(F \dashv G) \dashv (G \dashv H)
  \,.
\end{displaymath}
\end{prop}
This fact plays an important role in \hyperlink{LicataShulman}{Licata-Shulman, 5.1}. Relatedly, it also appears in the characterization of certain kinds of [[geometric morphism]] (e.g. the [[local geometric morphism|local]] ones) in terms of adjunctions in the 2-category [[Topos]].

It may be suggestive to denote this like so

\begin{displaymath}
\itexarray{
     F &\dashv& G
     \\
     \bot && \bot
     \\
     G &\dashv& H
  }
\end{displaymath}
such that the two adjoint pairs appear horizontally, while the second order adjunction between them runs vertically.

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\begin{note}
\label{GIsBicontinuous}\hypertarget{GIsBicontinuous}{}
The two adjunctions imply of course that $G$ preserves all [[limit]]s and [[colimit]]s that exist in $D$.

\end{note}
\begin{note}
\label{AdjointPairFromAdjointTriple}\hypertarget{AdjointPairFromAdjointTriple}{}
Every adjoint triple

\begin{displaymath}
(F \dashv G \dashv H) : C \to D
\end{displaymath}
gives rise to an [[adjunction|adjoint pair]]

\begin{displaymath}
(G F \dashv G H) : C \to C
\end{displaymath}
consisting of a [[monad]] $G F$ [[left adjoint]] to the [[comonad]] $G H$ on $C$;

as well as an adjoint pair

\begin{displaymath}
( F G \dashv H G ) : D \to D
   \,.
\end{displaymath}
\end{note}
See [[adjoint monad]] for more.

In general there is a duality (an anti[[equivalence of categories]]) between the category of monads having right adjoints and comonads having left adjoints. Note also that the [[algebra over a monad|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 (\hyperlink{SGL}{SGL}).)

\hypertarget{FullyFaithFulAdjointTriples}{}\subsubsection*{{Fully faithful adjoint triples}}\label{FullyFaithFulAdjointTriples}

\begin{prop}
\label{FullyFaithful}\hypertarget{FullyFaithful}{}
For an adjoint triple $F\dashv G\dashv H$ we have that $F$ is [[full and faithful functor|fully faithful]] precisely if $H$ is fully faithful.

\end{prop}
\begin{proof}
By a basic  of [[adjoint functors]] we have that

\begin{itemize}%
\item the [[left adjoint]] $F$ being full and faithful is equivalent to the [[unit of an adjunction|unit]] $Id \to G F$ being a [[natural isomorphism]];


\item the [[right adjoint]] $H$ being full and faithful is equivalent to the counit $G H \to Id$ being a [[natural isomorphism]].



\end{itemize}
Moreover, by note \ref{AdjointPairFromAdjointTriple} 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 functor]]s (for instance (\hyperlink{Elephant}{Elephant, lemma 1.1.1}) $G H$ is isomorphic to the identity precisely if it is so by the [[unit of an adjunction|adjunction unit]].

\end{proof}
The preceeding proposition is [[folklore]]; perhaps its earliest appearance in print is (\hyperlink{DyckhoffTholen}{DT, Lemma 1.3}). A slightly shorter proof is in (\hyperlink{KellyLawvere}{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-reduction|beta-reducing]] the above, slightly more abstract argument). It also appears in (\hyperlink{SGL}{SGL, Lemma 7.4.1}).

In the situation of Proposition \ref{FullyFaithful}, we say that $F\dashv G \dashv H$ is a \textbf{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 functor|faithful]]), but they do exhibit certain similar phenomena. In particular, we have the following.

\begin{prop}
\label{FinalLifts}\hypertarget{FinalLifts}{}
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 category|cocomplete]]. Then $G$ admits [[final lift|final lifts]] for [[small category|small]] $G$-structured [[sinks]].

\end{prop}
\begin{proof}
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.

\end{proof}
Thus, we can talk about objects of $D$ having the [[weak structure]] or [[strong structure]] induced by any small collection of maps.

\begin{cor}
\label{Fibration}\hypertarget{Fibration}{}
In the situation of Proposition \ref{FinalLifts}, $G$ is a ([[Street fibration|Street]]) [[Grothendieck fibration|opfibration]]. If it is also an [[isofibration]], then it is a Grothendieck opfibration.

\end{cor}
\begin{proof}
A final lift of a singleton sink is precisely an opcartesian arrow.

\end{proof}
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]].

\hypertarget{idempotent_adjoint_triples}{}\subsubsection*{{Idempotent adjoint triples}}\label{idempotent_adjoint_triples}

\begin{prop}
\label{Idempotent}\hypertarget{Idempotent}{}
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.

\end{prop}
\begin{proof}
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.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{special_cases}{}\subsubsection*{{Special cases}}\label{special_cases}

\begin{itemize}%
\item If one of the two [[adjoint pairs]] induced from an adjoint triple involving identities, then the other exhibits an \emph{[[adjoint cylinder]]} / \emph{[[unity of opposites]]}.


\item An adjoint triple $F\dashv G\dashv H$ is \textbf{Frobenius} if $F$ is naturally isomorphic to $H$. See [[Frobenius functor]].


\item An \emph{[[affine morphism]]} is an adjoint triple of functors in which the middle term is [[conservative functor|conservative]]. For example, any [[affine morphism of schemes]] induce an affine triples of functors among the categories of [[quasicoherent module]]s.


\item An adjoint triple of functors among $A_\infty$- or [[triangulated functor]]s with certain additional structure is called \textbf{spherical} . See e.g. (\hyperlink{Anno}{Anno}). The main examples come from [[Serre functor]]s in a [[Calabi-Yau category]] context.


\item 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 \emph{[[Wirthmüller context]]} or a \emph{[[Grothendieck context]]}, respectively.



\end{itemize}
\hypertarget{specific_examples}{}\subsubsection*{{Specific examples}}\label{specific_examples}

\begin{itemize}%
\item 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 [[coextension of scalars|right adjoint]]. This triple is affine in the above sense.


\item 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]].



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[adjoint quadruple]], [[adjoint string]]


\item [[cohesive topos]]


\item [[ambidextrous adjunction]]


\item [[affine morphism]], [[affine localization]]


\item [[Quillen adjoint triple]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Some remarks on adjoint triples are in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Remarks on punctual local connectedness} (\href{http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html}{tac})

\end{itemize}
The [[modal type theory]] of adjoint triples is discussed in

\begin{itemize}%
\item [[Dan Licata]], [[Mike Shulman]], \emph{Adjoint logic with a 2-category of modes} (\href{http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf}{pdf})

\end{itemize}
On spherical triples see

\begin{itemize}%
\item Rina Anno, \emph{Spherical functors}, (\href{http://arxiv.org/abs/0711.4409}{arxiv/0711.4409}).

\end{itemize}
Generalities are in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
Proofs of the folklore Proposition \ref{FullyFaithful} can be found in

\begin{itemize}%
\item Roy Dyckhoff and [[Walter Tholen]], ``Exponentiable morphisms, partial products, and pullback complements'', JPAA 49 (1987), 103--116.


\item [[G.M. Kelly]] and [[F.W. Lawvere]], ``On the complete lattice of essential localizations'', Bulletin de la Soci\'e{}t\'e{} Math\'e{}matique de Belgique, S\'e{}rie A, v. 41 no 2 (1989) 289--319.


\item [[Saunders Mac Lane]] and [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}, (1992) (Lemma 7.4.1).



\end{itemize}
Several lemmas concerning adjoint pairs and adjoint triples are included in

\begin{itemize}%
\item [[Alexander Rosenberg]], \emph{Noncommutative schemes}, Compos. Math. \textbf{112} (1998) 93--125, \href{http://dx.doi.org/10.1023/A:1000479824211}{doi}

\end{itemize}
together with geometric consequences. Note a somewhat nonstandard usage of terminology continuous functor (also flatness in the paper includes having right adjoint).

[[!redirects adjoint triples]] [[!redirects fully faithful adjoint triple]] [[!redirects fully faithful adjoint triples]] [[!redirects adjunction of adjunctions]] [[!redirects adjunction between adjunctions]]



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