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\begin{document}

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\section*{pushout}

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

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

[[!include category theory - contents]]

\hypertarget{infinitylimits}{}\paragraph*{{Infinity-limits}}\label{infinitylimits}

[[!include infinity-limits - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{pushouts_in_}{Pushouts in $Set$}\dotfill \pageref*{pushouts_in_} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{in_any_category}{In any category}\dotfill \pageref*{in_any_category} \linebreak
\noindent\hyperlink{in_a_quasitopos}{In a quasitopos}\dotfill \pageref*{in_a_quasitopos} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak


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

A \textbf{pushout} is an ubiquitous construction in [[category theory]] providing a [[colimit]] for the diagram $\bullet\leftarrow\bullet\rightarrow\bullet$. It is dual to the notion of a [[pullback]].

\hypertarget{pushouts_in_}{}\subsection*{{Pushouts in $Set$}}\label{pushouts_in_}

In the category [[Set]] a `pushout' is a quotient of the disjoint union of two sets. Given a diagram of sets and functions like this:

\begin{displaymath}
\itexarray{
 &&&&
     C
     &&&&
      \\
      & 
      && f \swarrow
       &
       & \searrow g
      && 
     \\
&& A &&&& B
  }
\end{displaymath}
the `pushout' of this diagram is the set $X$ obtained by taking the disjoint union $A + B$ and identifying $a \in A$ with $b \in B$ if there exists $x \in C$ such that $f(x) = a$ and $g(x) = b$ (and all identifications that follow to keep [[equality]] an [[equivalence relation]]).

This construction comes up, for example, when $C$ is the intersection of the sets $A$ and $B$, and $f$ and $g$ are the obvious inclusions. Then the pushout is just the union of $A$ and $B$.

Note that there are maps $i_A : A \to X$, $i_B : B \to X$ such that $i_A(a) = [a]$ and $i_B(b) = [b]$ respectively. These maps make this square commute:

\begin{displaymath}
\itexarray{
 &&&&
     C
     &&&&
      \\
      & 
      && f \swarrow
       &
       & \searrow g
      && 
     \\
&& A &&&& B
\\
      & 
      && {}_{i_A}\searrow
       &
       & \swarrow_{i_B}
      && 
     \\
&&&&
     X
     &&&&
  }
\end{displaymath}
In fact, the pushout is the [[universal property|universal]] solution to finding a [[commutative square]] like this. In other words, given \emph{any} commutative square

\begin{displaymath}
\itexarray{
 &&&&
     C
     &&&&
      \\
      & 
      && f \swarrow
       &
       & \searrow g
      && 
     \\
&& A &&&& B
\\
      & 
      && {}_{j_A}\searrow
       &
       & \swarrow_{j_B}
      && 
     \\
&&&&
     Y
     &&&&
  }
\end{displaymath}
there is a unique function $h: X \to Y$ such that

\begin{displaymath}
h i_A = j_A
\end{displaymath}
and

\begin{displaymath}
h i_B = j_B .
\end{displaymath}
Since this universal property expresses the concept of pushout purely arrow-theoretically, we can formulate it in any category. It is, in fact, a simple special case of a [[colimit]].

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

A \textbf{pushout} is a [[colimit]] of a [[diagram]] like this:

\begin{displaymath}
\itexarray{
 &&&&
     c
     &&&&
      \\
      & 
      && f \swarrow
       &
       & \searrow g
      && 
     \\
&& a &&&& b
  }
\end{displaymath}
Such a diagram is called a [[span]]. If the colimit exists, we obtain a [[commutative square]]

\begin{displaymath}
\itexarray{
 &&&&
     c
     &&&&
      \\
      & 
      && f \swarrow
       &
       & \searrow g
      && 
     \\
&& a &&&& b
\\
      & 
      && {}_{i_a}\searrow
       &
       & \swarrow_{i_b}
      && 
     \\
&&&&
     x
     &&&&
  }
\end{displaymath}
and the object $x$ is also called the \textbf{pushout}. It has the universal property already described above in the special case of the category $Set$.

Other terms: $x$ is a \textbf{cofibred coproduct} of $a$ and $b$, or (especially in [[algebraic categories]] when $f$ and $g$ are [[monomorphisms]]) a free product of $a$ and $b$ with $c$ \textbf{amalgamated}, or more simply an \textbf{amalgamation} (or \textbf{amalgam}) of $a$ and $b$.

The concept of pushout is a special case of the notion of \textbf{[[cofiber coproduct|wide pushout]]} (compare [[wide pullback]]), where one takes the colimit of a diagram which consists of a set of arrows $\{f_i: c \to a_i\}_{i \in I}$. Thus an ordinary pushout is the case where $I$ has cardinality $2$.

Note that the concept of pushout is dual to the concept of [[pullback]]: that is, a pushout in $C$ is the same as a pullback in $C^{op}$.

See [[pullback]] for more details.

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

\hypertarget{in_any_category}{}\subsubsection*{{In any category}}\label{in_any_category}

\begin{prop}
\label{PushoutsAsCoequalizers}\hypertarget{PushoutsAsCoequalizers}{}
\textbf{(pushouts as coequalizers)}

If [[coproduct]]s exist in some category, then the pushout

\begin{displaymath}
\itexarray{
     a &\stackrel{f}{\to} & b
     \\
     ^{\mathrlap{g}}\downarrow && \downarrow
     \\
     c & {\to}& b +_a c
  }
\end{displaymath}
is equivalently the [[coequalizer]]

\begin{displaymath}
a \stackrel{\overset{i_1 f}{\longrightarrow}}{\underset{i_2 g}{\longrightarrow}} b + c \to b +_a c
\end{displaymath}
of the two morphisms induced by $f$ and $g$ into the [[coproduct]] of $b$ with $c$.

\end{prop}
\begin{prop}
\label{PushoutPreservesEpimorphisms}\hypertarget{PushoutPreservesEpimorphisms}{}
\textbf{(pushouts preserves epimorphisms and isomorphisms)}

Pushouts preserve [[epimorphisms]] and [[isomorphisms]]:

If

\begin{displaymath}
\itexarray{
    a &\overset{f}{\longrightarrow}& b
    \\
    {}^{\mathllap{g}}\downarrow & & \downarrow^{f_\ast g}
    \\
    c &\underset{}{\longrightarrow}& d
  }
\end{displaymath}
is a pushout square in some category then:

\begin{enumerate}%
\item if $g$ is a [[epimorphism]] then $f_\ast g$ is an epimorphism;


\item if $g$ is an [[isomorphism]] then $f_\ast g$ is an isomorphism.



\end{enumerate}
\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{([[pasting law for pushouts]])}

Consider a [[commuting diagram]] of the following shape in any category:

\begin{displaymath}
\itexarray{
    x & \longrightarrow & y & \longrightarrow & z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    u & \longrightarrow & v & \longrightarrow & w
  }
\end{displaymath}
If the left square is a [[pushout]], then the total rectangle is a pushout if and only if the right square is a pushout.

\end{prop}
\begin{proof}
See the proof of the dual property for [[pullbacks]].

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
The converse implication does not hold: it may happen that the outer and the right square are pushouts, but not the left square.

\end{prop}
\begin{proof}
See the proof of the dual proposition for [[pullbacks]].

\end{proof}
\hypertarget{in_a_quasitopos}{}\subsubsection*{{In a quasitopos}}\label{in_a_quasitopos}

\begin{prop}
\label{PushoutOfStrongMonomorphismInQuasitopos}\hypertarget{PushoutOfStrongMonomorphismInQuasitopos}{}
\textbf{pushout of [[strong monomorphism]] in [[quasitopos]]}

Suppose that $(\mathrm{T},\mathcal{C})$ is either

\begin{itemize}%
\item ([[monomorphism]],[[topos]]), or
\item ([[strong monomorphism]],[[quasitopos]])

\end{itemize}
Suppose that

\begin{displaymath}
\itexarray{
O_{0,1} & \to & O_{1,1} \\
m \downarrow  &&\downarrow h  \\
O_{0,0} & \to & O_{1,0}
}
\end{displaymath}
is a [[commutative diagram]] in $\mathcal{C}$ such that

\begin{itemize}%
\item $m$ is $\mathrm{T}$ in $\mathcal{C}$
\item the diagram is a pushout in $\mathcal{C}$

\end{itemize}
Then

\begin{itemize}%
\item $h$ is $\mathrm{T}$ in $\mathcal{C}$
\item the diagram is a pullback in $\mathcal{C}$

\end{itemize}
\end{prop}
See at \emph{[[quasitopos]]} \href{quasitopos#PushoutOfStrongMonos}{this lemma}. Note that the result for quasitoposes immediately implies the result for toposes, since all monomorphisms $i: A \to B$ in a topos are [[regular monomorphism|regular]] ($i$ being the equalizer of the arrows $\chi_i, t \circ !: B \to \Omega$ in

\begin{displaymath}
\itexarray{
 & & 1 \\ 
 & \mathllap{!} \nearrow & \downarrow \mathrlap{t} \\ 
B & \underset{\chi_i}{\to} & \Omega
}
\end{displaymath}
where $\chi_i$ is the classifying map of $i$) and therefore strong.

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

\ldots{}

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[amalgamation property]]


\item [[wide pushout]]


\item [[coequalizer]]


\item [[colimit]]


\item [[pullback]]



\end{itemize}
[[!redirects pushout]] [[!redirects pushouts]]

[[!redirects cofibred coproduct]] [[!redirects cofibred coproducts]] [[!redirects cofibered coproduct]] [[!redirects cofibered coproducts]] [[!redirects cofiber coproduct]] [[!redirects cofiber coproducts]]

[[!redirects amalgamated free product]] [[!redirects amalgamated free products]] [[!redirects amalgamated coproduct]] [[!redirects amalgamated coproducts]] [[!redirects amalgamation]] [[!redirects amalgamations]] [[!redirects amalgam]] [[!redirects amalgams]]



\end{document}