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\begin{document}

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\section*{arity class}

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

\hypertarget{regular_and_exact_categories}{}\paragraph*{{Regular and Exact categories}}\label{regular_and_exact_categories}

[[!include regular and exact categories - contents]]

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

[[!include category theory - contents]]

\hypertarget{arity_classes}{}\section*{{Arity classes}}\label{arity_classes}

\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{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak


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

An \emph{arity class} is a class of [[cardinalities]] which is suitable to be the collection of [[arity|arities]] for the operations in an [[algebraic theory]].

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

An \textbf{arity class} is a [[class]] $\kappa$ of [[small set|small]] [[cardinalities]] such that

\begin{enumerate}%
\item $1\in\kappa$.


\item $\kappa$ is closed under indexed sums: if $\lambda\in\kappa$ and $\alpha: \lambda \to\kappa$, then $\sum_{i\in \lambda} \alpha(i)$ is also in $\kappa$.


\item $\kappa$ is closed under indexed decompositions: if $\lambda\in\kappa$ and $\sum_{i\in \lambda} \alpha(i)\in \kappa$, then each $\alpha(i)$ is also in $\kappa$.



\end{enumerate}
A [[set]] or [[family]] is called \textbf{$\kappa$-small} if its cardinality belongs to $\kappa$. A [[theory]] or other object with a collection of ``operations'' whose inputs are all $\kappa$-small is called \textbf{$\kappa$-ary}.

\begin{uremark}
By [[induction]], the second condition implies closure under iterated indexed sums, in the sense that for any $n\ge 2$, we have

\begin{displaymath}
\sum_{i_1\in\lambda_1} \; \sum_{i_2\in\lambda_2(i_1)} \cdots
  \sum_{i_{n-1} \in\lambda_{n-1}(i_1,\dots,i_{n-2})} \lambda_n(i_1,\dots,i_{n-1})
\end{displaymath}
is in $\kappa$ if all the $\lambda$`s are. The first condition may be regarded as the case $n=0$ of this (the case $n=1$ being just ``$\lambda\in\kappa$ iff $\lambda\in\kappa$'').

\end{uremark}
\begin{uremark}
An alternative, more category-theoretic, way to state the second and third conditions is that for any [[function]] $f:I\to J$, if ${|J|}\in\kappa$, then ${|I|}\in\kappa$ if and only if all fibers of $f$ are in $\kappa$.

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

\begin{itemize}%
\item The set $\{1\}$ is an arity class. A $\{1\}$-ary object is called \textbf{unary}.


\item The set $\{0,1\}$ is an arity class. A$\{0,1\}$-ary object is called \textbf{subunary}.


\item The set $\omega = \mathbb{N} = \{0,1,2,3\dots\}$ is an arity class. An $\omega$-ary object is called \textbf{finitary}.


\item For any [[regular cardinal]] $\kappa$, the set of all cardinalities strictly less than $\kappa$ is an arity class, which we abusively denote also by $\kappa$. The previous example $\omega$ is a special case of this, as is $\{0,1\}$ if we consider $2$ to be a regular cardinal.


\item In particular, if $\kappa$ is the ``size of the universe'' --- e.g., an [[inaccessible cardinal]] for which we have chosen to call sets of cardinality $\lt\kappa$ [[small set|small]], or literally the proper-class cardinality of the [[universe]], depending on how one thinks of it ---, then it is an arity class. In this case we call $\kappa$-ary objects \textbf{infinitary} or $\infty$-ary.



\end{itemize}
In [[classical mathematics]], these examples in fact exhaust \emph{all} arity classes. Classically, if $\lambda$ is any cardinal number strictly greater than $1$, then for any cardinal numbers $\mu\le \nu$, we can write $\nu$ as a $\lambda$-indexed sum containing $\mu$. Hence, if an arity class contains any cardinality $\gt 1$, it must be down-closed, and a down-closed arity class must arise from a regular cardinal.

In [[constructive mathematics]], however, not every arity class besides $\{1\}$ must be downward-closed, and not every downward-closed arity class must arise from a regular cardinal. Arguably, however, in constructive mathematics one should consider downward-closed arity classes instead of regular cardinals.

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

\begin{itemize}%
\item [[algebraic theory]]


\item [[∞-ary exact category]], [[∞-ary site]]


\item [[arity space]]



\end{itemize}
[[!redirects arity class]] [[!redirects arity classes]]

[[!redirects unary]] [[!redirects subunary]] [[!redirects finitary]] [[!redirects ∞-ary]] [[!redirects κ-ary]] [[!redirects kappa-ary]] [[!redirects infinitary]] [[!redirects infinity-ary]]



\end{document}