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\begin{document}

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\section*{adjoint quadruple}

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

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

[[!include category theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{canonical_natural_transformation}{Canonical natural transformation}\dotfill \pageref*{canonical_natural_transformation} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{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}

An \textbf{adjoint quadruple} is a sequence of three [[adjunctions]]

\begin{displaymath}
f_! \dashv f^* \dashv f_* \dashv f^!
\end{displaymath}
between a [[quadruple]] of [[morphisms]]. That is, it is an [[adjoint string]] of length 4.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Every adjoint quadruple

\begin{displaymath}
(f_! \dashv f^* \dashv f_* \dashv f^!)
  :
  C 
    \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\stackrel{\overset{f_*}{\to}}{\underset{f^!}{\leftarrow}}}}
  D
\end{displaymath}
induces an [[adjoint triple]] on $C$

\begin{displaymath}
(f^* f_! \dashv f^* f_* \dashv f^! f_*)
  :
  C \to C
  \,,
\end{displaymath}
(hence a [[monad]] [[left adjoint]] to a [[comonad]] left adjoint to a monad) and an adjoint triple

\begin{displaymath}
(f_! f^* \dashv f_* f^* \dashv f_* f^!) : D \to D
\end{displaymath}
on $D$.

Since moreover every [[adjoint triple]] $(F \dashv G \dashv H)$ induces an [[adjoint functor|adjoint pair]] $(G F \dashv G H)$ and an adjoint pair $(F G \dashv H G)$, the adjoint quadruple above induces four adjoint pairs, such as

\begin{displaymath}
(f^* f_* f^* f_! \dashv f^* f_* f^! f_*) : C \to C
  \,.
\end{displaymath}
$\,$

\hypertarget{canonical_natural_transformation}{}\subsubsection*{{Canonical natural transformation}}\label{canonical_natural_transformation}

Let

\begin{displaymath}
(p_! \dashv p^* \dashv p_*\dashv p^!) 
    \;\colon\; 
  \mathcal{E} 
    \longrightarrow 
  \mathcal{S}
\end{displaymath}
be an [[adjoint quadruple]] of [[adjoint functor]]s such that $p^*$ and $p^!$ are [[full and faithful functor]]s. We record some general properties of such a setup.

We write

\begin{displaymath}
\eta \;\colon\; id \to p^* p_!
\end{displaymath}
etc. for [[unit of an adjunction|units]] and

\begin{displaymath}
\epsilon \;\colon\; p_! p^* \to id
\end{displaymath}
etc. for counits.

\begin{prop}
\label{TheCanonicalMorphisms}\hypertarget{TheCanonicalMorphisms}{}
We have [[commuting diagrams]], [[natural transformation|natural]] in $X \in \mathcal{E}$, $S \in \mathcal{S}$

\begin{displaymath}
\itexarray{
    p_*X  &\underoverset{\simeq}{\epsilon_{p^* X}^{-1}}{\longrightarrow}& p_! p^* p_*X
    \\
    {}^{\mathllap{p_*(\eta_X)}}\downarrow 
    &\searrow^{\mathrlap{\theta_X}}& 
    \downarrow^{\mathrlap{p_!(\epsilon_X)}}
    \\
    p_* p^* p_! X &\stackrel{\eta_{p_!X}^{-1}}{\longrightarrow}& p_! X
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    p^* S &\stackrel{\eta_{p^* S}}{\longrightarrow}& p^! p_* p^* S
    \\
    {}^{\mathllap{p^* \epsilon_S^{-1}}}\downarrow 
    &\searrow^{\mathrlap{\phi_X}}& \downarrow^{\mathrlap{p^!(\eta_S^{-1})}}
    \\
    p^* p_* p^!S  &\stackrel{{\epsilon}_{p_!S }}{\longrightarrow}& p^!S
  }
  \,.
\end{displaymath}
where the diagonal morphisms

\begin{displaymath}
\theta_X : p_* X \to p_! X
\end{displaymath}
and

\begin{displaymath}
\phi_S : p^* S \to p^! S
\end{displaymath}
are defined to be the equal composites of the sides of these diagrams.

\end{prop}
This appears as (\hyperlink{Johnstone11}{Johnstone 11, lemma 2.1, corollary 2.2}).

\begin{prop}
\label{TheEpiAndTheMono}\hypertarget{TheEpiAndTheMono}{}
The following conditions are equivalent:

\begin{itemize}%
\item for all $X \in \mathcal{E}$ the morphism $\theta_X : p_*X \to p_! X$ is an [[epimorphism]];


\item for all $S \in \mathcal{S}$,, the morphism $\phi_S : p^*S \to p^! S$ is a [[monomorphism]];


\item $p_*$ is [[faithful functor|faithful]] on morphisms of the form $A \to p^* S$.



\end{itemize}
\end{prop}
This appears as (\hyperlink{Johnstone11}{Johnstone 11, lemma 2.3}).

\begin{proof}
By the above definition, $\phi_S$ is a [[monomorphism]] precisely if $\eta_{p^* S} : p^* S \to p^! p_* p^* S$ is. This in turn is so (see [[monomorphism]]) precisely if the first [[function]] in

\begin{displaymath}
\mathcal{E}(A,p^* X) 
     \stackrel{(\eta_{p^* X}) \circ (-)}{\longrightarrow} 
    \mathcal{E}(A, p^! p_* p^* S)
     \stackrel{\simeq}{\longrightarrow}
    \mathcal{S}(p_* A, p_* p^* S)
\end{displaymath}
and hence the composite is a monomorphism in [[Set]].

By definition of [[adjunct]] and using the $(p_* \dashv p^!)$-[[zig-zag identity]], this is equal to the action of $p_*$ on morphisms

\begin{displaymath}
(\eta_{p^* X}) \circ (-)  : 
  (A \to p^* S) \mapsto p_*(A \to p^* S)  
  \,.
\end{displaymath}
Similarly, by the above definition the morphism $\theta_X$ is an epimorphism precisely if $p_!(\epsilon_X) : p_! p^* p_* X \to p_! X$ is so, which is the case precisely if the top morphism in

\begin{displaymath}
\itexarray{
    \mathcal{S}(p_! X, S) 
      &\stackrel{(-) \circ p_!(\epsilon_X)}{\longrightarrow} &
    \mathcal{S}(p_! p^* p_* X, S)
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    && \mathcal{E}(p^* p_* X, p^* S)
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    \mathcal{E}(X, p^* S) &\stackrel{p_*}{\longrightarrow}& \mathcal{S}(p_* X, p_* p^* S)
  }
\end{displaymath}
and hence the bottom morphism is a monomorphism in [[Set]], where again the commutativity of this diagram follows from the definition of [[adjunct]] and the $(p_! \dashv p^*)$-[[zig-zag identity]].

\end{proof}
$\,$

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

\begin{example}
\label{KanExtensionOfAdjointPairIsAdjointQuadruple}\hypertarget{KanExtensionOfAdjointPairIsAdjointQuadruple}{}
\textbf{([[Kan extension]] of [[adjoint pair]] is [[adjoint quadruple]])}

For $\mathcal{V}$ a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] with all [[limits]] and [[colimits]], let $\mathcal{C}$, $\mathcal{D}$ be two [[small category|small]] $\mathcal{V}$-[[enriched categories]]and let

\begin{displaymath}
\mathcal{C}
    \underoverset
      {\underset{p}{\longrightarrow}}
      {\overset{q}{\longleftarrow}}
      {\bot}
  \mathcal{D}
\end{displaymath}
be a $\mathcal{V}$-[[enriched adjunction]]. Then there are $\mathcal{V}$-[[enriched natural isomorphisms]]

\begin{displaymath}
(q^{op})^\ast \;\simeq\; Lan_{p^{op}}
  \;\colon\;
  [\mathcal{C}^{op},\mathcal{V}]
    \longrightarrow
  [\mathcal{D}^{op},\mathcal{V}]
\end{displaymath}
\begin{displaymath}
(p^{op})^\ast \;\simeq\; Ran_{q^{op}}
  \;\colon\;
  [\mathcal{D}^{op},\mathcal{V}]
    \longrightarrow
  [\mathcal{C}^{op},\mathcal{V}]
\end{displaymath}
between the precomposition on [[enriched presheaves]] with one functor and the left/right [[Kan extension]] of the other.

By essential uniqueness of [[adjoint functors]], this means that the two [[Kan extension]] [[adjoint triples]] of $q$ and $p$

\begin{displaymath}
\itexarray{
    Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}}
    \\
    && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}}
  }
\end{displaymath}
merge into an [[adjoint quadruple]]

\begin{displaymath}
\itexarray{
    Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}}
  }
  \;\colon\;
  [\mathcal{C}^{op},\mathcal{V}]
    \leftrightarrow
  [\mathcal{D}^{op}, \mathcal{V}]
\end{displaymath}
\end{example}
\begin{proof}
For every [[enriched presheaf]] $F \;\colon\; \mathcal{C}^{op} \to \mathcal{V}$ we have a sequence of $\mathcal{V}$-[[enriched natural isomorphism]] as follows

\begin{displaymath}
\begin{aligned}
    (Lan_{p^{op}} F)(d)
      & \simeq
    \int^{ c \in \mathcal{C} } \mathcal{D}(d,p(c)) \otimes F(c)
    \\
    & \simeq
    \int^{ c \in \mathcal{C} } \mathcal{C}(q(d),c) \otimes F(c)
    \\
    & \simeq
    F(q(d))
    \\
    & = \left( (q^{op})^\ast F\right) (d)
    \,.    
  \end{aligned}
\end{displaymath}
Here the first step is the [[coend]]-formula for [[left Kan extension]] (\href{Kan+extension#PointwiseByCoEnds}{here}), the second step if the [[enriched adjunction]]-isomorphism for $q \dashv p$ and the third step is the [[co-Yoneda lemma]].

This shows the first statement, which, by essential uniqueness of adjoints, implies the following statements.

\end{proof}
\begin{itemize}%
\item For [[cohesive topos]] by definition the terminal [[geometric morphism]] extends to an adjoint quadruple.

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

\begin{itemize}%
\item [[adjunction]], [[adjoint functor]]


\item [[adjoint triple]], [[adjoint string]]



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

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Remarks on punctual local connectedness}, Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63. (\href{http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html}{tac})

\end{itemize}
[[!redirects adjoint quadruples]]



\end{document}