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\begin{document}

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\section*{parametric right adjoint}

\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{generic_morphisms}{Generic morphisms}\dotfill \pageref*{generic_morphisms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{free_categories}{Free categories}\dotfill \pageref*{free_categories} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Let $T\colon A\to B$ be a [[functor]] such that $A$ has a [[terminal object]] $1$. Then $T$ can canonically be factored as the composite

\begin{displaymath}
A \overset{T_1}{\to} B/T1 \overset{\Sigma_{T1}}{\to} B
\end{displaymath}
of $T$ applied to the [[slice category]] $A \simeq A/1$, followed by [[dependent sum]] (projection on the source).

We say that $T$ is a \textbf{parametric right adjoint}, or \textbf{p.r.a.}, if the functor $T_1$ is a [[right adjoint]]. Parametric right adjoints are also called \textbf{local right adjoints}.

A [[monad]] is called \textbf{p.r.a.} if its functor part is p.r.a. and moreover its unit and multiplication are [[cartesian natural transformation|cartesian]]. Thus in particular it is a [[cartesian monad]]. A p.r.a. monad is also called a \textbf{strongly cartesian monad}.

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

\begin{itemize}%
\item Since $\Sigma_{T1}$ [[created limit|creates]] [[connected limits]], if $T$ is p.r.a. then it [[preserved limit|preserves]] connected limits, and in particular preserves [[pullbacks]]. It follows that any p.r.a. monad is a [[cartesian monad]].


\item Conversely, an [[accessible functor]] between [[presheaf categories]] is p.r.a. if and only if it preserves [[connected limits]].


\item Any [[polynomial functor]] $\Sigma_h \Pi_g f^*$ is p.r.a., since then $T_1$ can be identified with $\Pi_g f^*$, which has the left adjoint $\Sigma_f \; g^*$.


\item If $E$ is a [[presheaf category]] and $T\colon E \to Set$ is p.r.a., then the [[comma category]] $Set/T$ (also called the [[Artin gluing]] in this context) is again a presheaf category. Conversely, if $Set/T$ is a presheaf category, then $T$ preserves connected limits, and thus is p.r.a. if it is accessible.



\end{itemize}
\hypertarget{generic_morphisms}{}\subsection*{{Generic morphisms}}\label{generic_morphisms}

Central to the theory of parametric right adjoints is the notion of \emph{$T$-generic} morphisms. For any functor $T$, a morphism $f\colon B\to T A$ is (strictly) \textbf{$T$-generic} if any commutative square of the following form:

\begin{displaymath}
\itexarray{B & \overset{\alpha}{\to} &T X \\
  ^f\downarrow && \downarrow^{T\gamma}\\
  T A& \underset{T \beta}{\to} & T Z}
\end{displaymath}
has a unique filler of the form $T\delta : T A \to T X$. A \textbf{generic factorization} of a map $f\colon B\to T A$ is a factorization

\begin{displaymath}
B \overset{g}{\to} T D \overset{T h}{\to} T A
\end{displaymath}
such that $g$ is $T$-generic. Note that by the definition of genericity, generic factorizations are unique whenever they exist. If $T$ is a monad and any map $B \to T A$ has a generic factorization, then there is an induced [[orthogonal factorization system]] on the [[Kleisli category]] of $T$ in which $T$-generic maps are the left class and the right class are the ``free'' maps, i.e. those which factor through the unit of $T$.

\begin{uprop}
A functor $T$ is a parametric right adjoint iff every map $B\to T A$ has a generic factorization.

\end{uprop}
\begin{proof}
This is Proposition 2.6 of \hyperlink{Weber08}{(Weber08)}.

\end{proof}
P.r.a. functors between presheaf categories have an especially nice form.

\begin{uprop}
A functor $T\colon [I^{op},Set] \to [J^{op},Set]$ between presheaf categories is p.r.a. iff any map $y(j)\to T 1$ has a generic factorization, where $y(j)$ is the representable presheaf on an object $j\in J$.

\end{uprop}
\begin{proof}
This is Proposition 2.10 of ``Familial 2-functors and parametric right adjoints.'' The ``only if'' direction is the previous proposition, while for the ``if'' direction, the given hypothesis allows us to define the functor

\begin{displaymath}
E_T \colon y/T1 \to [I^{op},Set]
\end{displaymath}
sending an object $(y(j) \to T 1)$ to the object occurring in its generic factorization. Note that $y/T1$ is equivalently the opposite of the [[category of elements]] of $T1$. The definition of genericity, along with the Yoneda lemma, then shows that

\begin{displaymath}
T(Z)(j) = \coprod_{x\in T1(j)} [I^{op},Set](E_T(x),Z)
\end{displaymath}
which preserves connected limits, since it is a coproduct of representables.

\end{proof}
In particular, a p.r.a. functor $T\colon [I^{op},Set] \to [J^{op},Set]$ is determined by giving the object $T1\in [J^{op},Set]$ together with the functor $E_T\colon y/T1 = el(T1)^{op} \to [I^{op},Set]$. We can think of $T1(j)$ as the setof all possible ``shapes'' which $T$ allows us to ``glue together'' to obtain an element of shape $j$, and $E_T$ as specifying exactly what each of those shapes looks like. Then the above formula for $T(Z)(j)$ says that we look at all possible shapes $x\in T1(j)$ we can glue to get something of shape $j$, and for each such $x$ we look at all the ``diagrams'' in $Z$ of the corresponding shape $E_T(x)$.

We can extract from this a description that is clearly a generalization of a [[polynomial functor]].

\begin{uprop}
A functor $T\colon [I^{op},Set] \to [J^{op},Set]$ between presheaf categories is p.r.a. iff when expressed in terms of [[discrete fibrations]], it is the composite

\begin{displaymath}
DFib/I \xrightarrow{d^*} DFib/E \xrightarrow{c_*} DFib/K \xrightarrow{p_!} DFib/J
\end{displaymath}
for a polynomial in $Cat$

\begin{displaymath}
I \xleftarrow{d} E \xrightarrow{c} K \xrightarrow{p} J
\end{displaymath}
where $p$ is a discrete fibration and $(d,c)$ is a [[two-sided discrete fibration]] (with in particular $d$ a discrete fibration and $c$ a discrete opfibration).

\end{uprop}
\begin{proof}
Let $p$ be the [[Grothendieck construction]] of $T1$, so that $K = el(T1)^{op}$, and $(d,c)$ the two-sided Grothendieck construction of $E_T\colon el(T1)^{op} \to [I^{op},Set]$ regarded as a profunctor from $K$ to $I$. The above formula tells us that $T = Lan_p \circ Hom(E_T,-)$, and when rewritten in terms of discrete fibrations this gives the above formula. More details are in Remark 2.12 of \hyperlink{Weber08}{(Weber08)}.

\end{proof}
That is, a p.r.a. functor between presheaf categories is the restriction to discrete fibrations of a certain kind of polynomial functor between slices of $Cat$. When $I$ and $J$ are discrete categories, then so are $K$ and $E$, so that p.r.a. functors between presheaf categories are a direct generalization of polynomial functors between slices of $Set$. But on the other hand, we can also say that polynomial functors between slices of \emph{Cat} are a direct generalization of p.r.a. functors between presheaf categories.

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

\hypertarget{free_categories}{}\subsubsection*{{Free categories}}\label{free_categories}

Consider the [[free category]] monad $T$ on the category $Quiv$ of [[quivers]], such that $T A$ is the quiver with the same objects as $A$ and whose arrows are finite composable strings of arrows in $A$.. Here $T 1$ is the monoid $\mathbb{N}$ regarded as a one-object category, and thus an object of $Quiv/T1$ is a quiver together with a natural number assigned to each edge. For any quiver $A$, the natural augmentation $T A \to T 1$ assigns to each composable string of arrows its length.

The left adjoint of this functor $T_1\colon Quiv \to Quiv/T1$ takes as input a quiver with natural number ``lengths'' assigned to each of its arrows, and creates a new quiver by gluing together a copy of the quiver $[n] = (0 \to 1 \to\dots \to n)$ (with no arrows other than those drawn) for each arrow of ``length'' $n$. Thus $T$ is a parametric right adjoint.

$Quiv$ is of course a presheaf category $[Q^{op},Set]$, where $Q$ is the category $0 \rightrightarrows 1$. The category $y/T1$, i.e. the opposite of the category of elements of $T1$, has objects $\mathbb{N} \sqcup \{\bot\}$ and nonidentity arrows $\bot \rightrightarrows n$ for all $n\in\mathbb{N}$. Finally, the functor $E_T \colon y/T1 \to Quiv$ sends $\bot$ to the quiver with one object and no arrows, and $n$ to the quiver $[n] = (0 \to 1 \to\dots \to n)$ described above.

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

\begin{itemize}%
\item [[Aurelio Carboni]] and [[Peter Johnstone]], \emph{Connected limits, familial representability and Artin glueing}, \href{http://www.ams.org/mathscinet-getitem?mr=1377312}{MR}


\item [[Mark Weber]], \emph{Generic morphisms, parametric representations, and weakly cartesian monads}, Theory and applications of categories, 13:191--234, 2004. \href{http://www.tac.mta.ca/tac/volumes/13/14/13-14abs.html}{link}


\item [[Mark Weber]], \emph{Familial 2-functors and parametric right adjoints}, 2008 \href{http://www.tac.mta.ca/tac/volumes/18/22/18-22abs.html}{link}



\end{itemize}
\begin{itemize}%
\item [[Clemens Berger]], [[Paul-André Melliès]], [[Mark Weber]], \emph{Monads with Arities and their Associated Theories} (2011) (\href{http://arxiv.org/abs/1101.3064}{arXiv:1101.3064})

\end{itemize}
[[!redirects parametric right adjoints]] [[!redirects p.r.a. functor]] [[!redirects p.r.a. functors]] [[!redirects p.r.a. monad]] [[!redirects p.r.a. monads]]

[[!redirects local right adjoint]] [[!redirects local right adjoints]] [[!redirects strongly cartesian monad]] [[!redirects strongly cartesian monads]]



\end{document}