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\begin{document}

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\section*{diagonal functor}

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

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

[[!include category theory - contents]]

\hypertarget{the_diagonal_functor}{}\section*{{The diagonal functor}}\label{the_diagonal_functor}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The diagonal functor is a [[categorification]] of the [[diagonal function]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $C$ be a [[category]]. The \textbf{(binary) diagonal functor of $C$} is the [[functor]] $\Delta\colon C \to C \times C$ given by $\Delta(x) = (x,x)$, regardless of whether $x$ is an [[object]] or an [[morphism|arrow]] of $C$.

More generally, let $J$ and $C$ be arbitrary categories. The \textbf{$J$-ary diagonal functor of $C$} is the functor $\Delta_J\colon C\to C^J$ sending each object $c$ to the [[constant functor]] $\Delta c$ (the functor having value $c$ for each object of $J$ and value $1_c$ for each arrow of $J$), and each arrow $f\colon c\to c'$ of $C$ to the [[natural transformation]] $\Delta f\colon \Delta c \stackrel{.}{\to} \Delta c'$ which has the same value $f$ at each object $j$ of $J$.

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

Since $C$ is $J$-cocomplete ($J$-complete) iff $\Delta$ has a left (right) [[adjoint functor|adjoint]], the general [[adjoint functor theorem]] may be used in some cases to prove [[cocomplete category|cocompleteness]] ([[complete category|completeness]]). For this to work, $\Delta$ must at least preserve small [[limits]] ([[colimits]]).

\begin{uprop}
Let $P$ and $C$ be arbitrary categories. Then $\Delta_P\colon C\to C^P$ preserves \emph{all} limits that exist in $C$.

\end{uprop}
\begin{proof}
First, recall that limits in [[functor categories]] are calculated pointwise. In some detail, if for an object $p\in \mathrm{obj}(P)$ we write $E_p:X^P\to X$ for the `'evaluate at $p$'` functor (with $E_p(H\colon P\to X)=H(p)$ and $E_p(\sigma\colon
H\stackrel{.}{\to} H')=\sigma_p\colon H(p)\to H'(p)$), then we have the following fact (Theorem V.3.1 on p. 115 of [[Categories Work]]): If $S\colon
J\to X^P$ is such that for each object $p$ of $P$, $E_p S\colon J\to X$ has a limiting cone $\tau_p\colon L(p)\stackrel{.}{\to} E_p S$, then there exists a unique functor $L$ with object function $p\mapsto L(p)$ such that $\tilde{\tau}=\{\tilde{\tau}_{j,p}\}$ with $\tilde{\tau}_{j,p}:=\tau_{p,j}$ is a cone $\tilde{\tau}\colon \Delta_J(L)\stackrel{.}{\to} S$; moreover, this $\tilde{\tau}$ is a limiting cone from $L\in \mathrm{obj}(X^P)$ to $S\colon J\to X^P$.

Back to the proof of the proposition, let $F\colon J\to C$ be a functor with a limiting cone $\nu\colon \Delta_J(\ell) \stackrel{.}{\to} F$. We would like to show that $\Delta_P\nu\colon \Delta_P\circ \bigl(\Delta_J(\ell)\bigr) \stackrel{.}{\to} \Delta_P\circ F$ is a limiting cone. Noting that $\Delta_P\circ \bigl(\Delta_J(\ell)\bigr)=\Delta_J(\Delta_P(\ell))$ (where the first $\Delta_J$ is $C\to C^J$ and the second is $C^P\to (C^P)^J$), the last cone may be written as $\Delta_P\nu\colon \Delta_J(\Delta_P(\ell))
\stackrel{.}{\to} \Delta_P\circ F$.

First, we note that for each object $p$ of $P$, $E_p\circ(\Delta_P\circ F)$ is just $F$, and therefore has the limiting cone $\nu\colon\ell \stackrel{.}{\to} F$ by assumption. Hence, it is clear that $\Delta_P\circ F$ has a limit, but we must verify that $\Delta_P\nu$ is a limiting cone.

One functor $P\to X$ with object function $p\mapsto \ell$ is just $\Delta_P(\ell)$. For this functor, we have our cone $\Delta_P\nu\colon \Delta_J(\Delta_P(\ell)) 
\stackrel{.}{\to} \Delta_P\circ F$. Since for all $j$ and $p$ we have $(\Delta_P\nu)_{j,p}=\nu_j=j\text{th component of the limiting cone of
}E_p\circ(\Delta_P\circ F)$, we are done by the theorem on pointwise limits.

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

\begin{itemize}%
\item [[diagonal morphism]]

\end{itemize}
[[!redirects diagonal functor]] [[!redirects diagonal functors]] [[!redirects diagonal functors preserve limits]]



\end{document}