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\begin{document}

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\section*{relative adjoint functor}

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

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

[[!include category theory - contents]]

\hypertarget{relative_adjoint_functors}{}\section*{{Relative adjoint functors}}\label{relative_adjoint_functors}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{via_homisomorphism}{Via hom-isomorphism}\dotfill \pageref*{via_homisomorphism} \linebreak
\noindent\hyperlink{via_absolute_lifting}{Via absolute lifting}\dotfill \pageref*{via_absolute_lifting} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{asymmetry}{asymmetry}\dotfill \pageref*{asymmetry} \linebreak
\noindent\hyperlink{unit_counit}{unit, counit}\dotfill \pageref*{unit_counit} \linebreak
\noindent\hyperlink{relative_monads_and_comonads}{relative monads and comonads}\dotfill \pageref*{relative_monads_and_comonads} \linebreak
\noindent\hyperlink{relative_adjointness_generalizes_adjointness}{relative adjointness generalizes adjointness}\dotfill \pageref*{relative_adjointness_generalizes_adjointness} \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{idea}{}\subsection*{{Idea}}\label{idea}

Relative adjoints with respect to a functor $J$ are a generalization of [[adjoint functor|adjoints]], where $J$ in the relative case plays the role of the identity in the standard setting: adjoints are the same as $Id$-relative adjoints.

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

\hypertarget{via_homisomorphism}{}\subsubsection*{{Via hom-isomorphism}}\label{via_homisomorphism}

Fix a functor $J\colon B \to D$. Then, a functor

\begin{displaymath}
R \colon C \to D
\end{displaymath}
has a \emph{left $J$-relative adjoint} (or $J$-left adjoint) if there is a functor

\begin{displaymath}
L \colon B \to C
\end{displaymath}
and a natural isomorphism

\begin{displaymath}
Hom_C(L(-),-) \simeq Hom_D(J(-),R(-))
\end{displaymath}
Dually, $L \colon C \to D$ has a \emph{$J$-right adjoint} $R \colon B \to C$ if there's a natural isomorphism

\begin{displaymath}
Hom_D(L(-), J(-)) \simeq Hom_C(-, R(-))
\end{displaymath}
\begin{itemize}%
\item $L {\,\,}_J\!\dashv R$ stands for $L$ being the $J$-left adjoint of $R$
\item $L \dashv_J R$ stands for $R$ being the $J$-right adjoint of $L$

\end{itemize}
\hypertarget{via_absolute_lifting}{}\subsubsection*{{Via absolute lifting}}\label{via_absolute_lifting}

Just as with regular adjoints, relative adjoints can be defined in a more conceptual way in terms of \emph{absolute [[Kan lift|liftings]]}. We have

\begin{enumerate}%
\item $L {\,\,}_J\!\dashv R$ if $L = \operatorname{Lift}_R J$, and this left lifting is \emph{absolute}
\item $L \dashv_J R$ if $R = \operatorname{Rift}_L J$, and this right lifting is \emph{absolute}

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

\hypertarget{asymmetry}{}\subsubsection*{{asymmetry}}\label{asymmetry}

The most important difference with regular adjunctions is the asymmetry of the concept. First, for $L {\,\,}_J\!\dashv R$ it makes no sense to ask for $R \dashv_J L$ (domains and comodomains do not typecheck). And secondly, and more importantly:

\begin{itemize}%
\item \textbf{$L$ is $J$-left adjoint to $R$:} $R$ \emph{determines} $L$
\item \textbf{$R$ is $J$-right adjoint to $L$:} $L$ \emph{determines} $R$

\end{itemize}
(this is obvious from the definition in terms of liftings). Because of this, even if most of the properties of adjunctions have a generalization to the relative setting, they do that in a one-sided way.

\hypertarget{unit_counit}{}\subsubsection*{{unit, counit}}\label{unit_counit}

Asymmetry manifests itself here:

\begin{enumerate}%
\item $L {\,\,}_J\!\dashv R$ yields a $J$-relative \emph{unit} 2-cell $\iota\colon J \to R L$
\item while $L \dashv_J R$ gives a $J$-relative \emph{counit} $\epsilon\colon L R \to J$

\end{enumerate}
with no naturally available counterpart for them in each case.

These 2-cells are directly provided by the definition in terms of liftings, as the universal 2-cells

\begin{itemize}%
\item $\iota\colon J \to R L$ given by $L = \operatorname{Lift}_R J$
\item $\epsilon\colon L R \to J$ given by $R = \operatorname{Rift}_L J$

\end{itemize}
Alternatively, and just as with regular adjunctions, their components can be obtained from the natural hom-isomorphism: in the unit case, evaluating at $Lb$ we get a bijection

\begin{displaymath}
Hom_C(Lb, Lb) \simeq Hom_D(Jb, RLb)
\end{displaymath}
and

\begin{displaymath}
\iota_b \colon J b \to R L b
\end{displaymath}
is given by evaluating at $1_{Lb}$ the aforementioned bijection. A completely analogous procedure yields a description of the counit for $L \dashv_J R$.

\hypertarget{relative_monads_and_comonads}{}\subsubsection*{{relative monads and comonads}}\label{relative_monads_and_comonads}

Just as adjunctions give rise to [[monad|monads]] and [[comonad|comonads]], for relative adjoints

\begin{enumerate}%
\item If $L {\,\,}_J\!\dashv R$, then $RL$ is a [[relative monad]]
\item If $L \dashv_J R$, then $LR$ is a [[relative comonad]]

\end{enumerate}
with relative units and counits as above, respectively.

There are also relative analogues of [[Eilenberg-Moore category|Eilenberg-Moore]] and [[Kleisli category|Kleisli]] categories for these.

\hypertarget{relative_adjointness_generalizes_adjointness}{}\subsubsection*{{relative adjointness generalizes adjointness}}\label{relative_adjointness_generalizes_adjointness}

The concept of relative adjoint functors is a generalization of the concept of adjoint functors: if a functor $R\colon C\to D$ has a left adjoint in the usual sense, then it also has a $J$-left adjoint for $J=id_D$.

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

\begin{description}

\item[\textbf{fully faithful functors}] A functor $F: A \to B$ is fully faithful iff it is representably fully faithful iff $1_A = \operatorname{Lift}_F F$, and this lifting is absolute. Thus, $F$ fully faithful can be expressed as

\begin{displaymath}
1 {\,\,}_F\!\dashv F
\end{displaymath}
 

\item[\textbf{partially defined adjoints}] As remarked in the [[adjoint functor|local definition of adjoint functor]], given a functor

\begin{displaymath}
L \colon C \to D
\end{displaymath}
it may happen that $Hom_D(L(-),d)$ is [[representable functor|representable]] only for \emph{some} $d$, but not for all of them. In that case, taking

\begin{displaymath}
J \colon B \to D
\end{displaymath}
be the inclusion of the [[full subcategory|full subcategory]] determined by $Hom_D(L(-),d)$ representable, and defining $R \colon B \to C$ accordingly, we have

\begin{displaymath}
L \dashv_J R
\end{displaymath}
This can be specialized to situations such as a category having \emph{some} but not all [[limit|limits]] of some kind, partially defined [[Kan extension|Kan extensions]], etc. See also [[free object]].

 

\item[\textbf{nerves}] Take $A$ a locally small category, and $F\colon A \to B$ a locally left-small functor (one for which $B(Fa,b)$ is always small). The \emph{$A$-nerve} induced by $F$ is the functor

\begin{displaymath}
N_F \colon B \to \mathbf{Set}^{A^{\operatorname{op}}}
\end{displaymath}
given by $N_F(b)(a) = A(Fa,b)$. It is a fundamental fact that $F = \operatorname{Lift}_{N_F} y_A$ and this lifting is \emph{absolute}; or, in relative adjoint notation, $F {\,\,}_{y_A}\!\dashv N_F$. The universal 2-cell $\iota\colon y_A \to N_F F$ is given by the action of $F$ on morphisms:

\begin{displaymath}
\iota_a \colon y_A a \to (N_F F)(a)
\end{displaymath}
at $a' \colon A$ is

\begin{displaymath}
F_{a,a'}\colon A(a,a') \to B(Fa, Fa')
\end{displaymath}
Note that when specialized to $F = 1_A$, this reduces to the [[yoneda lemma|Yoneda lemma]]: first $N_{1_A} \simeq y_A$, and then $1_A = \operatorname{Lift}_{y_A} y_A$ absolute in hom-isomorphism terms reads:

\begin{displaymath}
A(x,y) \simeq \mathbf{Set}^{A^{\operatorname{op}}}(y_A x, y_A y)
\end{displaymath}
One of the axioms of a [[yoneda structure|Yoneda structure]] on a 2-category abstract over this situation, by requiring the existence of $F$-nerves with respect to yoneda embeddings such that the 1-cell $F$ is an absolute left lifting as above; see \hyperlink{Weber2007}{Weber} or \hyperlink{StreetWalters1978}{Street--Walters} .

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

\begin{itemize}%
\item [[relative monad]]

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

\begin{itemize}%
\item [[Thorsten Altenkirch]], James Chapman and Tarmo Uustalu, \emph{Monads need not be endofunctors} In: Ong L. (eds) Foundations of Software Science and Computational Structures. FoSSaCS 2010. Lecture Notes in Computer Science, vol 6014. Springer, Berlin, Heidelberg, arXiv:1412.7148 cs.PL ,\href{https://arxiv.org/pdf/1412.7148.pdf}{pdf}


\item [[Ross Street]], Bob Walters - \emph{Yoneda structures on 2-categories}, Journal of Algebra, Volume 50, Issue 2, February 1978, Pages 350-379, \href{http://www.mendeley.com/research/yoneda-structures-2categories/}{article at mendeley}


\item F. Ulmer, \emph{Properties of dense and relative adjoint functors}, Journal of Algebra, Volume 8, Issue 1, 1968, Pages 77-95, \href{http://www.sciencedirect.com/science/article/pii/0021869368900367}{pdf}


\item [[Mark Weber]] - \emph{Yoneda structures from 2-toposes}, Appl Categor Struct (2007) 15: 259. doi:10.1007/s10485-007-9079-2, \href{https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes}{pdf}



\end{itemize}
[[!redirects relative adjoint functors]] [[!redirects relative right adjoint]] [[!redirects relative left adjoint]] [[!redirects relative adjoint]] [[!redirects relative adjoints]] [[!redirects relative adjunction]] [[!redirects relative adjunctions]]



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