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\begin{document}

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\section*{disjoint coproduct}

\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_a_category}{In a category}\dotfill \pageref*{in_a_category} \linebreak
\noindent\hyperlink{in_a_bicategory}{In a bicategory}\dotfill \pageref*{in_a_bicategory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{characterization_of_extensivity_and_of_sheaf_toposes}{Characterization of extensivity and of sheaf toposes}\dotfill \pageref*{characterization_of_extensivity_and_of_sheaf_toposes} \linebreak
\noindent\hyperlink{in_coherent_categories}{In coherent categories}\dotfill \pageref*{in_coherent_categories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The notion of \emph{disjoint coproduct} is a generalization to arbitrary [[categories]] of that of \emph{[[disjoint union]]} of sets.

One says that a [[coproduct]] $X + Y$ of two [[objects]] $X, Y$ in a [[category]] $\mathcal{C}$ is \emph{disjoint} if the [[intersection]] of $X$ with $Y$ in $X \coprod Y$ is [[initial object|empty]]. In this case one writes $X \coprod Y \coloneqq X + Y$ for the coproduct and speaks of the \emph{[[disjoint union]]} of $X$ with $Y$.

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

\hypertarget{in_a_category}{}\subsubsection*{{In a category}}\label{in_a_category}

A [[coproduct]] $a+b$ in a [[category]] is \textbf{disjoint} if

\begin{enumerate}%
\item the coprojections $a\to a+b$ and $b\to a+b$ are [[monomorphism|monic]], and


\item their [[intersection]] is an [[initial object]].



\end{enumerate}
Equivalently, this means we have [[pullback]] squares

\begin{displaymath}
\itexarray{ a & \to & a &&&
b & \to & b &&&
0 & \to & b\\
\downarrow && \downarrow &&&
\downarrow && \downarrow &&&
\downarrow && \downarrow \\
a & \to & a+b &&&
b & \to & a+b &&&
a & \to & a+b}
\end{displaymath}
An arbitrary coproduct $\coprod_i a_i$ is disjoint if each coprojection $a_i\to \coprod_i a_i$ is monic and the intersection of any two is initial. Note that every 0-ary coproduct (that, is initial object) is disjoint.

\hypertarget{in_a_bicategory}{}\subsubsection*{{In a bicategory}}\label{in_a_bicategory}

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

\hypertarget{characterization_of_extensivity_and_of_sheaf_toposes}{}\subsubsection*{{Characterization of extensivity and of sheaf toposes}}\label{characterization_of_extensivity_and_of_sheaf_toposes}

A category having all finitary disjoint coproducts is half of the condition for a category to be [[extensive category|extensive]].

Having all small disjoint coproducts is one of the conditions in [[Giraud's theorem]] characterizing [[sheaf toposes]].

\hypertarget{in_coherent_categories}{}\subsubsection*{{In coherent categories}}\label{in_coherent_categories}

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a [[coherent category]]. If $X, Y \hookrightarrow Z$ are two [[subobjects]] of some [[object]] $Z \in \mathcal{C}$ and are disjoint, in that their [[intersection]] in $Z$ is [[initial object|empty]], $X \cap Y \simeq\emptyset$, then their [[union]] $X \cup Y$ is their (disjoint) [[coproduct]].

\end{prop}
This apears as (\hyperlink{Johnstone}{Johnstone, cor. A1.4.4}).

\begin{defn}
\label{PositiveCategory}\hypertarget{PositiveCategory}{}
A coherent category in which all coproducts are disjoint is also called a \textbf{[[positive category|positive coherent category]]}.

\end{defn}
(\hyperlink{Johnstone}{Johnstone, p. 34})

\begin{example}
\label{}\hypertarget{}{}
Every [[extensive category]] is in particular positive, by definition.

\end{example}
In a positive coherent category, every morphism into a coproduct factors through the coproduct coprojections:

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a postive coherent category, def. \ref{PositiveCategory}, and let $f \colon A \to X \coprod Y$ be a [[morphism]]. Then the two [[subobjects]] $f^*(X) \hookrightarrow A$ and $f^*(Y) \hookrightarrow Y$ of $A$, being the [[pullbacks]] in

\begin{displaymath}
\itexarray{
    f^* (X) &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{i_X}}
    \\
    A &\stackrel{f}{\to}& X \coprod Y
  }
  \;\;\;\;
  \itexarray{
    f^* (Y) &\to& Y
    \\
    \downarrow && \downarrow^{\mathrlap{i_Y}}
    \\
    A &\stackrel{f}{\to}& X \coprod Y
  }
\end{displaymath}
are disjoint in $A$ and $A$ is their disjoint coproduct

\begin{displaymath}
A \simeq f^*(X) \coprod f^*(Y)
  \,.
\end{displaymath}
\end{prop}
This appears in (\hyperlink{Johnstone}{Johnstone, p. 34}).

\begin{remark}
\label{}\hypertarget{}{}
This means that if $A \in \mathcal{C}$ itself is indecomposable in that it is not a coproduct of two objects in a non-trivial way, for instance if $\mathcal{C}$ is an [[extensive category]] and $A \in \mathcal{C}$ is a [[connected object]], then every morphism $A \to X \coprod Y$ into a disjoint coproduct factors through one of the two canonical inclusions.

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

\begin{itemize}%
\item [[disjoint subset]]

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

For instance page 34 in section A1.4.4 in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
[[!redirects disjoint coproduct]] [[!redirects disjoint coproducts]]



\end{document}