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\begin{document}

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\section*{connected limit}

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

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{construction_from_pullbacks_and_equalizers}{Construction from pullbacks and equalizers}\dotfill \pageref*{construction_from_pullbacks_and_equalizers} \linebreak
\noindent\hyperlink{preservation_from_wide_pullbacks}{Preservation from wide pullbacks}\dotfill \pageref*{preservation_from_wide_pullbacks} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{connected limit} is a [[limit]] over a [[connected category]]. Similarly, a \textbf{connected colimit} is a colimit over a connected category.

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

The following are all connected limits:

\begin{itemize}%
\item [[pullbacks]]


\item [[wide pullbacks]]


\item [[equalizers]]


\item [[cofiltered limits]]


\item [[split idempotents]]



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

\hypertarget{construction_from_pullbacks_and_equalizers}{}\subsubsection*{{Construction from pullbacks and equalizers}}\label{construction_from_pullbacks_and_equalizers}

\begin{theorem}
\label{PbEqToFinConnLim}\hypertarget{PbEqToFinConnLim}{}
If a category $C$ has [[pullbacks]] and [[equalizers]], then it has all [[finite limit|finite]] connected limits.

\end{theorem}
\begin{proof}
Let $I$ be a finite connected category and $F\colon I\to C$ a functor. Since $I$ is connected, it is inhabited; choose some object $x_0\in I$. For each object $x\in I$, let $\ell(x)$ be the length of the shortest [[zigzag]] from $x_0$ to $x$. Now order the objects of $I$ as $x_0, x_1, \dots, x_n$ such that for all $i$ we have $\ell(x_i) \le \ell(x_{i+i})$.

Now we inductively define objects $P_i\in C$, for $0\le i\le n$, with projections $p_{i j}\colon P_i \to F(x_j)$ for $j\le i$. We begin with $P_0 = x_0$ and $p_{0 0}= 1_{x_0}$. Assuming $P_i$ and $p_{i j}$ defined, choose a zigzag from $x_0$ to $x_{i+1}$ of minimal length, say

\begin{displaymath}
x_0 \leftrightarrow y_1 \leftrightarrow \dots \leftrightarrow y_k \leftrightarrow x_{i+1}.
\end{displaymath}
By our choice of the ordering of the objects of $I$, we have $y_k = x_j$ for some $j\le i$, and thus we have $q = p_{i j}\colon P_i \to y_k$.

If the final arrow $y_k \leftrightarrow x_{i+1}$ in the zigzag is directed as $y_k \to x_{i+1}$, then let $P_{i+1} = P_i$, let $p_{i+1, i+1}$ be the composite $P_i \xrightarrow{q} F(y_k) \to F(x_{i+1})$, and keep the other $p_{i j}$ unchanged. On the other hand, if $y_k \leftrightarrow x_{i+1}$ is directed as $y_k \leftarrow x_{i+1}$, let $P_{i+1}$ be the pullback

\begin{displaymath}
\itexarray{
& P_{i+1} & \to & P_i \\
^{p_{i+1,i+1}} & \downarrow & & \downarrow^q \\
& F(x_{i+1}) & \to & F(y_k)
}
\end{displaymath}
and define $p_{i+1,j}$ for $j\le i$ by composition with $P_{i+1}\to P_i$.

At the end of this procedure, we have an object $P_n$ with projections $p_{n,j}\colon P_n \to F(x_j)$ for all objects $x_j\in I$. Now we order the morphisms in $I$ as $g_0,\dots,g_m$ and define, inductively, an object $Q_i$ with a morphism $q_i\colon Q_i \to P_n$. We begin with $Q_0 = P_n$ and $q_0 = 1_{P_n}$. Given $Q_i$ and $q_i$, we let $e\colon Q_{i+1} \to Q_{i}$ be the equalizer of $F(g_{i+1}) \circ p_{n,?} \circ q_i$ and $p_{n,?} \circ q_i$, where the ?s denote the indices of the objects that are the source and target of $g_{i+1}$. We then set $q_{i+1} = q_i \circ e$.

At the end of this procedure, we have an object $Q_{m+1}$ together with a cone over the diagram $F$, which is easily verified to be a limit of $F$.

\end{proof}
Similarly, arbitrary connected limits may be built from [[wide pullbacks]] and equalizers.

\begin{theorem}
\label{internalsum}\hypertarget{internalsum}{}
Let $C$ be (finitely) complete, and let $X$ be an object of $C$. Then the forgetful functor

\begin{displaymath}
\sum_X: C/X \to C
\end{displaymath}
preserves and reflects (finite) connected limits.

\end{theorem}
\begin{proof}
The wide pullback of the diagram $f_i: X_i \to X$ where $f_i = 1_X$ for all $i$ is clearly $X$, as is the equalizer of the pair of identity arrows. Since the product functor $C \times C \to C$ preserves arbitrary limits, we see that

\begin{displaymath}
X \times lim (c_i \stackrel{g_i}{\to} c) = (lim X_i \stackrel{f_i = 1_X}{\to} X) \times (lim c_i \stackrel{g_i}{\to} c) = lim X \times c_i \stackrel{1 \times g_i}{\to} X \times c,
\end{displaymath}
i.e., $X \times -$ preserves wide pullbacks. The same line of argument shows $X \times -$ preserves equalizers, so $X \times -$ preserves connected limits.

The functor $X \times -$ carries a canonical [[comonad]] structure, whose category of coalgebras is $C/X$. The forgetful functor $C/X \to C$ is comonadic, and thus preserves and reflects any class of limits preserved by the comonad $X \times -$. Thus $\sum_X: C/X \to C$ preserves and reflects all connected limits.

\end{proof}
\hypertarget{preservation_from_wide_pullbacks}{}\subsubsection*{{Preservation from wide pullbacks}}\label{preservation_from_wide_pullbacks}

Recall that a [[wide pullback]] is a limit over a [[diagram]] whose underlying shape is the [[poset]] obtained by [[free functor|freely]] adjoining a [[top element|terminal element]] to a [[set|discrete poset]], which is certainly connected.

It is not true that if $C$ has wide pullbacks then it has connected limits. The [[saturated class of limits|saturation]] of the class of wide pullbacks is the class of connected and ``simply connected'' limits (limits over categories $C$ whose groupoid reflection $\Pi_1(C)$ is trivial).

However, the following is true.

\begin{theorem}
\label{PreservesWidePbToConnected}\hypertarget{PreservesWidePbToConnected}{}
Let $C$ be a complete category, and let $D$ be [[locally small category|locally small]]. Then a functor $G\colon C \to D$ preserves connected limits if and only if it preserves wide pullbacks.

\end{theorem}
\begin{proof}
The forward direction is clear since wide pullbacks are examples of connected limits. Now suppose $G\colon C \to D$ preserves wide pullbacks. Then

\begin{equation}
C \stackrel{G}{\to} D \stackrel{hom(d, -)}{\to} Set
\label{Gwithhom}\end{equation}
preserves wide pullbacks for every object $d$ of $D$. Put $I = hom(d, G 1)$. The underlying functor

\begin{displaymath}
\sum\colon Set/I \to Set
\end{displaymath}
reflects and preserves connected limits and in particular wide pullbacks, so that the evident lift

\begin{displaymath}
\hom(d, G-)\colon C \to Set/I
\end{displaymath}
preserves wide pullbacks. It also preserves the terminal object, hence by \href{/nlab/show/wide+pullback#WidePbToComplete}{this proposition} it preserves arbitrary limits. Therefore the composite

\begin{displaymath}
C \stackrel{\hom(d, G-)}{\to} Set/I \stackrel{\sum}{\to} Set
\end{displaymath}
preserves connected limits, for every object $d$. Since this is the same composite as in \eqref{Gwithhom}, and since the representables $\hom(d, -)$ jointly reflect arbitrary limits, we conclude that $G$ preserves connected limits.

\end{proof}
The analogous argument works for finite limits. In particular, for $C$ finitely complete, a functor $G\colon C \to D$ that preserves pullbacks also preserves [[equalizers]].

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[parametric right adjoint]]


\item [[connected category]]


\item [[wide pullback]]


\item [[wide pushout]]



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

\begin{itemize}%
\item [[Robert Paré]], \emph{Simply connected limits}. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, \href{http://math.ca/10.4153/CJM-1990-038-6}{CMS}

\end{itemize}
[[!redirects connected limit]] [[!redirects connected limits]]



\end{document}