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\begin{document}

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\section*{free diagram}

\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{Exposition}{Exposition}\dotfill \pageref*{Exposition} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \emph{free diagram} in a [[category]] $\mathcal{C}$ is a particularly simple special case of the general concept of a \emph{[[diagram]]} $X_\bullet \;\colon\; \mathcal{I} \to \mathcal{C}$, namely the case where the shape $\mathcal{I}$ of the diagram is a [[free category]].

Many important types of [[limits]] and [[colimits]] are over free diagrams, for instance [[products]]/[[coproducts]], [[equalizers]]/[[coequalizers]], [[pullbacks]]/[[pushouts]], [[sequential limits]]/[[sequential colimits]].

Due to the simplicity of the concept of free diagrams, these types of [[limits]] and [[colimits]] may be discussed in a very low-brow way, without even making the concept of \emph{[[category]]} and \emph{[[functor]]} explicit. For this see the \emph{\hyperlink{Exposition}{Exposition}} below.

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

Recall that

\begin{defn}
\label{Diagram}\hypertarget{Diagram}{}
\textbf{([[diagram]])}

For $\mathcal{C}$ a [[category]], then a \emph{[[diagram]] in $\mathcal{C}$} is

\begin{enumerate}%
\item a [[small category]] $\mathcal{I}$ , the \emph{shape} of the diagram;


\item a [[functor]] $X_\bullet \;\colon\; \mathcal{I} \to \mathcal{C}$.



\end{enumerate}
\end{defn}
\begin{defn}
\label{FreeCategory}\hypertarget{FreeCategory}{}
\textbf{([[free category]])}

There is a [[free-forgetful adjunction]]

\begin{displaymath}
Cat
    \underoverset
      {\underset{Underlying}{\longrightarrow}}
      {\overset{Free}{\longleftarrow}}
      {\bot}
  DirGraph
\end{displaymath}
between the [[1-categories]] of [[categories]] and that of [[directed graphs]].

A \emph{[[free category]]} is one in the [[image]] of this [[left adjoint]] functor $Free \colon DirGraph \to Cat$ (sometimes called a ``[[path category]]'').

\end{defn}
\begin{defn}
\label{FreeDiagram}\hypertarget{FreeDiagram}{}
\textbf{(free diagram)}

A \emph{[[free diagram]]} in a [[category]] $\mathcal{C}$ is a [[diagram]] in $\mathcal{C}$ (def. \ref{Diagram}) whose shape is a [[free category]] (def. \ref{FreeCategory}).

In other words, a free diagram in $\mathcal{C}$ is

\begin{enumerate}%
\item a [[directed graph]] $I$;


\item a [[functor]] of the form $X_\bullet \;\colon\; Free(I) \to \mathcal{C}$.



\end{enumerate}
\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
Types of free diagrams that are commonly encountered in practice, as well as the names of the [[limits]]/[[colimits]] over them are shown in the following table

[[!include free diagrams -- table]]

\end{example}
\hypertarget{Exposition}{}\subsection*{{Exposition}}\label{Exposition}

We give an exposition of free diagrams, and their [[cones]] and [[limits]], intentionally avoiding abstract category-theoretic language, expressing everything just in components. See also at \emph{[[limits and colimits by example]]}.

For concreteness, we speak only of diagrams of [[sets]] and of [[topological spaces]] in the following:

\begin{defn}
\label{Diagram}\hypertarget{Diagram}{}
\textbf{([[free diagram]] of [[sets]]/[[topological spaces]])}

A \emph{[[free diagram]]} $X_\bullet$ of [[sets]] or of [[topological spaces]] is

\begin{enumerate}%
\item a [[set]] $\{ X_i \}_{i \in I}$ of [[sets]] or of [[topological spaces]], respectively;


\item for every [[pair]] $(i,j) \in I \times I$ of labels, a [[set]] $\{ X_i \overset{ f_\alpha }{\longrightarrow} X_j\}_{\alpha \in I_{i,j}}$ of [[functions]] of of [[continuous functions]], respectively, between these.



\end{enumerate}
\end{defn}
\begin{example}
\label{DiscreteDiagram}\hypertarget{DiscreteDiagram}{}
\textbf{([[discrete category|discrete]] [[diagram]] and [[empty diagram]])}

Let $I$ be any [[set]], and for each $(i,j) \in I \times I$ let $I_{i,j} = \emptyset$ be the [[empty set]].

The corresponding [[free diagrams]] (def. \ref{Diagram}) are simply a set of sets/topological spaces with no specified (continuous) functions between them. This is called a \emph{[[discrete category|discrete]] [[diagram]]}.

For example for $I = \{1,2,3\}$ the set with 3-elements, then such a diagram looks like this:

\begin{displaymath}
X_1 \phantom{AAA} X_2 \phantom{AAA} X_3
  \,.
\end{displaymath}
Notice that here the index set may be [[empty set]], $I = \emptyset$, in which case the corresponding diagram consists of no data. This is also called the \emph{[[empty diagram]]}.

\end{example}
\begin{defn}
\label{ParallelMorphisms}\hypertarget{ParallelMorphisms}{}
\textbf{([[parallel morphisms]] [[diagram]])}

Let $I = \{a, b\}$ be the [[set]] with two elements, and consider the sets

\begin{displaymath}
I_{i,j}
    \;\coloneqq\;
  \left\{
    \itexarray{
      \{ 1,2 \} & \vert & (i = a) \,\text{and}\, (j = b)
      \\
      \emptyset & \vert & \text{otherwise}
    }
  \right\}
  \,.
\end{displaymath}
The corresponding [[free diagrams]] (def. \ref{Diagram}) are called \emph{[[pairs of parallel morphisms]]}. They may be depicted like so:

\begin{displaymath}
X_a
    \underoverset
      {\underset{f_2}{\longrightarrow}}
      {\overset{f_1}{\longrightarrow}}
      {\phantom{AAAAA}}
  X_b
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{SpanDiagram}\hypertarget{SpanDiagram}{}
\textbf{([[span]] and [[cospan]] [[diagram]])}

Let $I = \{a,b,c\}$ the set with three elements, and set

\begin{displaymath}
I_{i ,j}
   =
  \left\{
    \itexarray{
      \{f_1\} & \vert \, (i = c) \,\text{and}\, (j = a)
      \\
      \{f_2\} & \vert \, (i = c) \,\text{and}\, (j = b)
      \\
      \emptyset & \vert \, \text{otherwise}
    }
  \right.
\end{displaymath}
The corresponding [[free diagrams]] (def. \ref{Diagram}) look like so:

\begin{displaymath}
\itexarray{
    && X_c
    \\
    & {}^{\mathllap{f_1}}\swarrow && \searrow^{\mathrlap{f_2}}
    \\
    X_a && && X_b
  }
  \,.
\end{displaymath}
These are called \emph{[[span]] [[diagrams]]}.

Similary, there is the \emph{[[cospan]]} diagram of the form

\begin{displaymath}
\itexarray{
    && X_c
    \\
    & {}^{\mathllap{f_1}}\nearrow && \nwarrow^{\mathrlap{f_2}}
    \\
    X_a && && X_b
  }
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
\textbf{([[tower]] [[diagram]])}

Let $I = \mathbb{N}$ be the set of [[natural numbers]] and consider

\begin{displaymath}
I_{i,j}
   \;\coloneqq\;
  \left\{
    \itexarray{
      \{f_{i,j}\} & \vert & i \leq j
      \\
      \emptyset & \vert & \text{otherwise}
    }
  \right.
\end{displaymath}
The corresponding [[free diagrams]] (def. \ref{Diagram}) are called \emph{[[tower]] [[diagrams]]}. They look as follows:

\begin{displaymath}
X_0
     \overset{\phantom{A}f_{0,1} \phantom{A} }{\longrightarrow}
   X_1
     \overset{\phantom{A} f_{1,2} \phantom{A} }{\longrightarrow}
   X_2
     \overset{\phantom{A} f_{2,3} \phantom{A} }{\longrightarrow}
   X_3
     \overset{}{\longrightarrow}
   \cdots
   \,.
\end{displaymath}
Similarly there are co-tower diagram

\begin{displaymath}
X_0
     \overset{\phantom{A} f_{0,1} \phantom{A} }{\longleftarrow}
   X_1
     \overset{\phantom{A} f_{1,2} \phantom{A}}{\longleftarrow}
   X_2
     \overset{\phantom{A} f_{2,3} \phantom{A}}{\longleftarrow}
   X_3
     \overset{}{\longleftarrow}
   \cdots
   \,.
\end{displaymath}
\end{example}
$\,$

\begin{defn}
\label{Cone}\hypertarget{Cone}{}
\textbf{([[cone]] over a [[free diagram]])}

Consider a [[free diagram]] of sets or of topological spaces (def. \ref{Diagram})

\begin{displaymath}
X_\bullet
    \,=\,
  \left\{
    X_i \overset{f_\alpha}{\longrightarrow} X_j
  \right\}_{i,j \in I, \alpha \in I_{i,j}}
  \,.
\end{displaymath}
Then

\begin{enumerate}%
\item a \emph{[[cone]]} over this diagram is

\begin{enumerate}%
\item a [[set]] or [[topological space]] $\tilde X$ (called the \emph{tip} of the cone);


\item for each $i \in I$ a [[function]] or [[continuous function]] $\tilde X \overset{p_i}{\longrightarrow} X_i$



\end{enumerate}
such that

\begin{itemize}%
\item for all $(i,j) \in I \times I$ and all $\alpha \in I_{i,j}$ then the condition

\begin{displaymath}
f_{\alpha} \circ p_i = p_j
\end{displaymath}
holds, which we depict as follows:

\begin{displaymath}
\itexarray{
    && \tilde X
    \\
    & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}}
    \\
    X_i
  && \underset{f_\alpha}{\longrightarrow} &&
    X_j
  }
\end{displaymath}


\end{itemize}

\item a \emph{[[co-cone]]} over this diagram is

\begin{enumerate}%
\item a set or topological space $\tilde X$ (called the \emph{tip} of the co-cone);


\item for each $i \in I$ a function or continuous function $q_i \colon X_i \longrightarrow \tilde X$;



\end{enumerate}
such that

\begin{itemize}%
\item for all $(i,j) \in I \times I$ and all $\alpha \in I_{i,j}$ then the condition

\begin{displaymath}
q_j \circ f_{\alpha} = q_i
\end{displaymath}
holds, which we depict as follows:

\begin{displaymath}
\itexarray{
    X_i && \overset{f_\alpha}{\longrightarrow} && X_j
    \\
    & {}_{\mathllap{q_i}}\searrow && \swarrow_{\mathrlap{q_j}}
    \\
    && \tilde X
  }
  \,.
\end{displaymath}


\end{itemize}


\end{enumerate}
\end{defn}
\begin{example}
\label{ConeIncarnationOfSolutionsToEquations}\hypertarget{ConeIncarnationOfSolutionsToEquations}{}
\textbf{([[solutions]] to [[equations]] are [[cones]])}

Let $f,g \colon \mathbb{R} \to \mathbb{R}$ be two [[functions]] from the [[real numbers]] to themselves, and consider the corresponding [[parallel morphism]] [[diagram]] of sets (example \ref{ParallelMorphisms}):

\begin{displaymath}
\mathbb{R}
    \underoverset
      {\underset{f_2}{\longrightarrow}}
      {\overset{f_1}{\longrightarrow}}
      {\phantom{AAAAA}}
  \mathbb{R}
  \,.
\end{displaymath}
Then a [[cone]] (def. \ref{Cone}) over this free diagram with tip the [[singleton]] set $\ast$ is a \emph{[[solution]]} to the [[equation]] $f(x) = g(x)$

\begin{displaymath}
\itexarray{
    && \ast
    \\
    & {}^{\mathllap{const_x}}\swarrow && \searrow^{\mathrlap{const_y}}
    \\
    \mathbb{R}
      &&
      \underoverset
        {\underset{f_2}{\longrightarrow}}
        {\overset{f_1}{\longrightarrow}}
        {\phantom{AAAAA}}
      &&
    \mathbb{R}
  }
  \,.
\end{displaymath}
Namely the components of the cone are two functions of the form

\begin{displaymath}
cont_x, const_y \;\colon\; \ast \to \mathbb{R}
\end{displaymath}
hence equivalently two [[real numbers]], and the conditions on these are

\begin{displaymath}
f_1 \circ const_x = const_y
  \phantom{AAAA}
  f_2 \circ const_x = const_y
  \,.
\end{displaymath}
\end{example}
\begin{defn}
\label{LimitingCone}\hypertarget{LimitingCone}{}
\textbf{([[limit|limiting cone]] over a [[diagram]])}

Consider a [[free diagram]] of sets or of topological spaces (def. \ref{Diagram}):

\begin{displaymath}
\left\{
    X_i \overset{f_\alpha}{\longrightarrow} X_j
  \right\}_{i,j \in I, \alpha \in I_{i,j}}
  \,.
\end{displaymath}
Then

\begin{enumerate}%
\item its \emph{[[limit|limiting cone]]} (or just \emph{[[limit]]} for short, also ``[[inverse limit]]'', for historical reasons) is [[generalized the|the]] [[cone]]

\begin{displaymath}
\left\{
  \itexarray{
    && \underset{\longleftarrow}{\lim}_k X_k
    \\
    & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}}
    \\
    X_i
      && \underset{f_\alpha}{\longrightarrow} &&
    X_j
  }
  \right\}
\end{displaymath}
over this diagram (def. \ref{Cone}) which is \emph{[[universal property|universal]]} among all possible cones, in that for

\begin{displaymath}
\left\{
  \itexarray{
    && \tilde X
    \\
    & {}^{\mathllap{p'_i}}\swarrow && \searrow^{\mathrlap{p'_j}}
    \\
    X_i
      && \underset{f_\alpha}{\longrightarrow} &&
    X_j
  }
  \right\}
\end{displaymath}
any other [[cone]], then there is a unique function or continuous function, respectively

\begin{displaymath}
\phi \;\colon\; \tilde X \overset{}{\longrightarrow} \underset{\longrightarrow}{\lim}_i X_i
\end{displaymath}
that factors the given cone through the limiting cone, in that for all $i \in I$ then

\begin{displaymath}
p'_i = p_i \circ \phi
\end{displaymath}
which we depict as follows:

\begin{displaymath}
\itexarray{
    \tilde X
    \\
    {}^{\mathllap{ \exists !\, \phi}}\downarrow & \searrow^{\mathrlap{p'_i}}
    \\
    \underset{\longrightarrow}{\lim}_i X_i
    &\underset{p_i}{\longrightarrow}&
    X_i
  }
\end{displaymath}

\item its \emph{[[colimit|colimiting cocone]]} (or just \emph{[[colimit]]} for short, also ``[[direct limit]]'', for historical reasons) is [[generalized the|the]] [[cocone]]

\begin{displaymath}
\left\{
  \itexarray{
    X_i
      && \overset{f_\alpha}{\longrightarrow} &&
    X_j
    \\
    & {}^{\mathllap{q_i}}\searrow && \swarrow^{\mathrlap{q_j}}
    \\
    \\
    && \underset{\longrightarrow}{\lim}_i X_i
  }
  \right\}
\end{displaymath}
under this diagram (def. \ref{Cone}) which is \emph{[[universal property|universal]]} among all possible co-cones, in that it has the property that for

\begin{displaymath}
\left\{
  \itexarray{
    X_i
      && \overset{f_\alpha}{\longrightarrow} &&
    X_j
    \\
    & {}^{\mathllap{q'_i}}\searrow && \swarrow_{\mathrlap{q'_j}}
    \\
    && \tilde X
  }
  \right\}
\end{displaymath}
any other [[cocone]], then there is a unique function or continuous function, respectively

\begin{displaymath}
\phi \;\colon\;  \underset{\longrightarrow}{\lim}_i X_i \overset{}{\longrightarrow} \tilde X
\end{displaymath}
that factors the given co-cone through the co-limiting cocone, in that for all $i \in I$ then

\begin{displaymath}
q'_i = \phi \circ q_i
\end{displaymath}
which we depict as follows:

\begin{displaymath}
\itexarray{
    X_i
    &\overset{q_i}{\longrightarrow}&
    \underset{\longrightarrow}{\lim}_i X_i
    \\
    & {}_{q'_i}\searrow & \downarrow^{\mathrlap{\exists ! \phi}}
    \\
    && \tilde X
  }
\end{displaymath}


\end{enumerate}
\end{defn}
$\,$

All the limits and colimits over the free diagram in the above list of examples have special names:

[[!include free diagrams -- table]]

\begin{example}
\label{TerminalInitialObject}\hypertarget{TerminalInitialObject}{}
\textbf{([[initial object]] and [[terminal object]])}

Consider the [[empty diagram]] (def. \ref{DiscreteDiagram}).

\begin{enumerate}%
\item A [[cone]] over the empty diagram is just an object $X$, with no further structure or condition. The [[universal property]] of the [[limit]] ``ast$'' over the empty diagram is hence that
for every object$X$, there is a unique map of the form$X $\backslash$to $\backslash$ast$. Such an object$$\backslash$ast\$ is called a \emph{[[terminal object]]}.


\item A [[co.cone]] over the empty diagram is just an object $X$, with no further structure or condition. The [[universal property]] of the [[colimit]] ``$'' over the empty diagram is hence that
for every object$X$, there is a unique map of the form$0 $\backslash$to X$. Such an object$$\backslash$ast\$ is called a \emph{[[initial object]]}.



\end{enumerate}
\end{example}
\begin{example}
\label{CoProduct}\hypertarget{CoProduct}{}
\textbf{([[Cartesian product]] and [[coproduct]])}

Let $\{X_i\}_{i \in I}$ be a [[discrete category|discrete]] [[diagram]] (example \ref{DiscreteDiagram}), i.e. just a set of objects.

\begin{enumerate}%
\item The [[limit]] over this diagram is called the \emph{[[Cartesian product]]}, denoted $\underset{i \in I}{\prod} X_i$;


\item The [[colimit]] over this diagram is called the \emph{[[coproducts]]}, denoted $\underset{i \in I}{\coprod} X_i$.



\end{enumerate}
\end{example}
\begin{example}
\label{Equalizer}\hypertarget{Equalizer}{}
\textbf{([[equalizer]])}

Let

\begin{displaymath}
X_1
    \underoverset
      {\underset{\phantom{AA}f_2\phantom{AA}}{\longrightarrow}}
      {\overset{\phantom{AA}f_1\phantom{AA}}{\longrightarrow}}
      {}
  X_2
\end{displaymath}
be a [[free diagram]] of the shape ``[[pair of parallel morphisms]]'' (example \ref{ParallelMorphisms}).

A [[limit]] over this diagram according to def. \ref{LimitingCone} is also called the \emph{[[equalizer]]} of the maps $f_1$ and $f_2$. This is a set or topological space $eq(f_1,f_2)$ equipped with a map $eq(f_1,f_2) \overset{p_1}{\longrightarrow} X_1$, so that $f_1 \circ p_1 = f_2 \circ p_1$ and such that if $Y \to X_1$ is any other map with this property

\begin{displaymath}
\itexarray{
    && Y
    \\
    && \downarrow & \searrow
    \\
    eq(f_1,f_2)
      &\overset{p_1}{\longrightarrow}&
    X_1
      &
        \underoverset
          {\underset{\phantom{AA}f_2\phantom{AA}}{\longrightarrow}}
          {\overset{\phantom{AA}f_1\phantom{AA}}{\longrightarrow}}
          {}
      &
    X_2
  }
\end{displaymath}
then there is a unique factorization through the equalizer:

\begin{displaymath}
\itexarray{
    && Y
    \\
    &{}^{\mathllap{\exists !}}\swarrow& \downarrow & \searrow
    \\
    eq(f_1,f_2)
      &\overset{p_1}{\longrightarrow}&
    X_1
      &
        \underoverset
          {\underset{f_2}{\longrightarrow}}
          {\overset{f_1}{\longrightarrow}}
          {}
      &
    X_2
  }
  \,.
\end{displaymath}
In example \ref{ConeIncarnationOfSolutionsToEquations} we have seen that a cone over such a pair of parallel morphisms is a \emph{[[solution]]} to the equation $f_1(x) = f_2(x)$.

The equalizer above is the \emph{space of all solutions} of this equation.

\end{example}
\begin{example}
\label{Pushout}\hypertarget{Pushout}{}
\textbf{([[pullback]]/[[fiber product]] and [[coproduct]])}

Consider a [[cospan]] [[diagram]] (example \ref{SpanDiagram})

\begin{displaymath}
\itexarray{
     && Y
     \\
     && \downarrow^{\mathrlap{f}}
     \\
     X
       &\underset{g}{\longrightarrow}&
     Z
  }
  \,.
\end{displaymath}
The [[limit]] over this diagram is also called the \emph{[[fiber product]]} of $X$ with $Y$ over $Z$, and denoted $X \underset{Z}{\times}Y$. Thought of as equipped with the projection map to $X$, this is also called the \emph{[[pullback]]} of $f$ along $g$

\begin{displaymath}
\itexarray{
     X \underset{X}{\times} Z
       &\longrightarrow&
     Y
     \\
     \downarrow &(pb)& \downarrow^{\mathrlap{f}}
     \\
     X
       &\underset{g}{\longrightarrow}&
     Z
  }
  \,.
\end{displaymath}
[[formal duality|Dually]], consider a [[span]] [[diagram]] (example \ref{SpanDiagram})

\begin{displaymath}
\itexarray{
    Z
      &\overset{g}{\longrightarrow}&
    Y
    \\
    {}^{\mathllap{f}}\downarrow
    \\
    X
  }
\end{displaymath}
The [[colimit]] over this diagram is also called the [[pushout]] of $f$ along $g$, denoted $X \underset{Z}{\sqcup}Y$:

\begin{displaymath}
\itexarray{
    Z
      &\overset{g}{\longrightarrow}&
    Y
    \\
    {}^{\mathllap{f}}\downarrow &(po)& \downarrow
    \\
    X &\longrightarrow& X \underset{Z}{\sqcup} Y
  }
\end{displaymath}
\end{example}
$\,$

Here is a more explicit description of the limiting cone over a diagram of sets:

\begin{prop}
\label{SetLimits}\hypertarget{SetLimits}{}
\textbf{(\href{limits+and+colimits+by+example#limcoliminset}{limits and colimits of sets})}

Let $\left\{ X_i \overset{f_\alpha}{\longrightarrow} X_j \right\}_{i,j \in I, \alpha \in I_{i,j}}$ be a [[free diagram]] of sets (def. \ref{Diagram}). Then

\begin{enumerate}%
\item its [[limit|limit cone]] (def. \ref{LimitingCone}) is given by the following [[subset]] of the [[Cartesian product]] $\underset{i \in I}{\prod} X_i$ of all the [[sets]] $X_i$ appearing in the diagram

\begin{displaymath}
\underset{\longleftarrow}{\lim}_i X_i
  \,\overset{\phantom{AAA}}{\hookrightarrow}\,
  \underset{i \in I}{\prod} X_i
\end{displaymath}
on those [[tuples]] of elements which match the [[graphs]] of the functions appearing in the diagram:

\begin{displaymath}
\underset{\longleftarrow}{\lim}_{i} X_i
  \;\simeq\;
  \left\{
    (x_i)_{i \in I}
    \,\vert\,
    \underset{ {i,j \in I} \atop { \alpha \in I_{i,j} } }{\forall}
    \left(
      f_{\alpha}(x_i) = x_j
    \right)
  \right\}
\end{displaymath}
and the projection functions are $p_i \colon (x_j)_{j \in I} \mapsto x_i$.


\item its [[colimit|colimiting co-cone]] (def. \ref{LimitingCone}) is given by the [[quotient set]] of the [[disjoint union]] $\underset{i \in I}{\sqcup} X_i$ of all the [[sets]] $X_i$ appearing in the diagram

\begin{displaymath}
\underset{i \in I}{\sqcup} X_i
    \,\overset{\phantom{AAA}}{\longrightarrow}\,
  \underset{\longrightarrow}{\lim}_{i \in I} X_i
\end{displaymath}
with respect to the [[equivalence relation]] which is generated from the [[graphs]] of the functions in the diagram:

\begin{displaymath}
\underset{\longrightarrow}{\lim}_i X_i
  \;\simeq\;
  \left(
    \underset{i \in I}{\sqcup} X_i
  \right)/
  \left(
    (x \sim x')
      \Leftrightarrow
    \left(
      \underset{ {i,j \in I} \atop { \alpha \in I_{i,j} } }{\exists}
      \left(
    f_\alpha(x) = x'
      \right)
    \right)
  \right)
\end{displaymath}
and the injection functions are the evident maps to [[equivalence classes]]:

\begin{displaymath}
q_i \;\colon\; x_i \mapsto [x_i]
  \,.
\end{displaymath}


\end{enumerate}
\end{prop}
\begin{proof}
We dicuss the proof of the first case. The second is directly analogous.

First observe that indeed, by consturction, the projection maps $p_i$ as given do make a cone over the free diagram, by the very nature of the relation that is imposed on the tuples:

\begin{displaymath}
\itexarray{
     &&
     \left\{
       (x_k)_{k \in I}
       \,\vert\,
       \underset{ {i,j \in I} \atop { \alpha \in I_{i,j} } }{\forall}
       \left(
         f_{\alpha}(x_i) = x_j
       \right)
     \right\}
    \\
    & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}}
    \\
    X_i && \underset{f_\alpha}{\longrightarrow} && X_j
  }
  \,.
\end{displaymath}
We need to show that this is universal, in that any other cone over the free diagram factors universally through it. First consider the case that the tip of a give cone is a singleton:

\begin{displaymath}
\itexarray{
    && \ast
    \\
    & {}^{\mathllap{p'_i}}\swarrow && \searrow^{\mathrlap{p'_j}}
    \\
    X_i
      && \underset{f_\alpha}{\longrightarrow} &&
    X_j
  }
  \,.
\end{displaymath}
This is hence equivalently for each $i \in I$ an element $x'_i \in X_i$, such that for all $i, j \in I$ and $\alpha \in I_{i,j}$ then $f_\alpha(x'_i) = x'_j$. But this is precisely the relation used in the construction of the limit above and hence there is a unique map

\begin{displaymath}
\ast
    \longrightarrow
     \left\{
       (x_k)_{k \in I}
       \,\vert\,
       \underset{ {i,j \in I} \atop { \alpha \in I_{i,j} } }{\forall}
       \left(
         f_{\alpha}(x_i) = x_j
       \right)
     \right\}
\end{displaymath}
such that for all $i \in I$ we have

\begin{displaymath}
\itexarray{
    \ast
    \\
    \downarrow & \searrow^{\mathrlap{p'_i}}
    \\
     \left\{
       (x_k)_{k \in I}
       \,\vert\,
       \underset{ {i,j \in I} \atop { \alpha \in I_{i,j} } }{\forall}
       \left(
         f_{\alpha}(x_i) = x_j
       \right)
     \right\}
     &\underset{p_i}{\longrightarrow}&
     X_i
  }
\end{displaymath}
namely that map is the one that picks the element $(x'_i)_{i \in I}$.

This shows that every cone with tip a singleton factors uniquely through the claimed limiting cone. But then for a cone with tip an arbitrary set $Y$, this same argument applies to all the single elements of $Y$.

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

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


\item [[commuting diagram]]


\item [[internal diagram]]



\end{itemize}
[[!redirects free diagrams]]



\end{document}