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\begin{document}

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\section*{disjoint union}

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

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

[[!include category theory - contents]]

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

[[!include infinity-limits - contents]]

\hypertarget{disjoint_unions}{}\section*{{Disjoint unions}}\label{disjoint_unions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{internal_version}{Internal version}\dotfill \pageref*{internal_version} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

The \emph{disjoint union} is a [[coproduct]] in [[Set]], the category of [[sets]].

In a general [[category]] [[coproducts]] need not have the expected disjointness property of those in [[Set]]. If they do they are called \emph{[[disjoint coproducts]]}.

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

Given any family $(A_i)_{i:I}$ of sets, the (external) \textbf{disjoint union} $\biguplus_i A_i$ (also written $\sum_i A_i$, $\coprod_i A_i$, etc) of the family is the set of all (ordered) pairs $(i,a)$ with $i$ in the index set $I$ and $a$ in $A_i$.

As stated, the type of the second element of such a pair depends on the first element, which is natural in [[dependent type theory]] (see at \emph{[[dependent sum type]]}) and no problem for [[material set theory]], but it may be ill formed in a [[structural set theory]] or in some forms of [[type theory]], especially those based on the [[internal language]] of [[topos theory]]. Alternatively, one may define $\biguplus_i A_i$ to be the set of those elements $x$ of the [[cartesian product]] $\prod_i \mathcal{P}A_i$ of the [[power sets]] such that there is exactly one index $j$ such that $x_j$ is [[inhabited set|inhabited]] and that $x_j$ is a [[singleton]]. If you're trying to be [[predicative mathematics|predicative]] too, then you may need to simply adopt the existence of disjoint unions as an axiom (the \textbf{axiom of disjoint unions}) in your [[foundations]], stating the following facts about it.

There is a natural [[injection]] $A_j \to \biguplus_i A_i$ (mapping $a$ to $(j,a)$, or mapping $a$ to $(i \mapsto \{a \;|\; i = j\})$) for each index $j$. Conversely, for each element $x$ of $\biguplus_i A_i$, there is a unique index $j$ and such that $x$ is in the [[image]] of the injection from $A_j$. It is common to treat $A_j$ as a [[subset]] of $\biguplus_i A_i$; so if no confusion can result (in particular, when the notation for an element of $A_j$ always makes the ambient set clear), one often suppresses the index in the notation for an element of the disjoint union.

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

Given sets $A$ and $B$, the disjoint union of the binary family $(A,B)$ is written $A \uplus B$ (also $A + B$, $A \amalg B$, etc); its elements may be written (if care is needed) as $(0,a)$ and $(1,b)$, $(1,a)$ and $(2,b)$, $\iota{a}$ and $\kappa{b}$, and in many other styles.

Given sets $A_1$ through $A_n$, the disjoint union of the $n$-ary family $(A_1,\ldots,A_n)$ is written $\biguplus_{i=1}^n A_i$ (or similarly); its elements may be written (if care is needed) as $(i,a)$ for $1 \leq i \leq n$ and $a \in A_i$.

Given sets $A_1$, $A_2$, etc, the disjoint union of the countably infinitary family $(A_1,A_2,\ldots)$ is written $\biguplus_{i=1}^\infty A_i$ (or similarly); its elements may be written (if care is needed) as $(i,a)$ for $i$ a natural number and $a \in A_i$.

Given a set $A$, the disjoint union of the unary family $(A)$ may be identified with $A$ itself; that is, we identify $(i,a)$ for the unique index $i$ with $a$.

The disjoint union of the empty family $()$ is [[empty set|empty]]; it has no elements.

\hypertarget{internal_version}{}\subsection*{{Internal version}}\label{internal_version}

(This is internal in the sense of `internal direct sum', not [[internalization]]. For that, just see [[coproduct]].)

If a family $(A_i)_{i: I}$ of [[subsets]] of a given set $X$ are all pairwise disjoint (that is, $A_i \cap A_j$ has an element only if $i = j$, for any indices $i$ and $j$), then the [[union]] $\bigcup_i A_i$ is [[natural isomorphism|naturally bijective]] with the (external) disjoint union defined above. Conversely, given an external disjoint union $\biguplus_i A_i$, each $A_j$ may be identified with a subset of $\biguplus_i A_i$ (as explained above); these subsets are all pairwise disjoint, and their union is the entire disjoint union.

Accordingly, a union of pairwise disjoint subsets may be called an \textbf{internal disjoint union}. (Compare the internal vs external notions of [[direct sum]].)

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

\begin{itemize}%
\item [[countable unions of countable sets are countable]]

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

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


\item [[dependent sum]], [[dependent sum type]]


\item [[Cartesian product]]


\item [[disjoint union topological space]]



\end{itemize}
category: foundational axiom

[[!redirects disjoint union]] [[!redirects disjoint unions]]

[[!redirects external disjoint union]] [[!redirects external disjoint unions]]

[[!redirects internal disjoint union]] [[!redirects internal disjoint unions]]

[[!redirects axiom of disjoint unions]]



\end{document}