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\begin{document}

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\section*{class function (set theory)}

\begin{quote}%
This entry is about the term of ``class function'' as it is used in set-theoretical contexts. For the notion of the same name in [[algebra]] see at \emph{[[class function]]}.


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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{example_in_category_theory}{Example in Category Theory}\dotfill \pageref*{example_in_category_theory} \linebreak
\noindent\hyperlink{example_in_set_theory}{Example in Set Theory}\dotfill \pageref*{example_in_set_theory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Recall that in [[set theory]] a [[function]] $f$ from a [[set]] $S_1$ to a set $S_2$ may be encoded in terms of a [[relation]] on the [[Cartesian product]] $S_1 \times S_2$ of the two sets, namely the subset $R \subset S^1 \times S^2$ with $R = \{  (x,y) \vert y = f(x)\}$ (the [[graph]] of the function).

This concept has an evident generalization to the case where $S_1$ and $S_2$ are allowed to be proper classes. In this case one speaks of \emph{class functions}.

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

A class function is a [[class]] $R$ which is a [[relation]] with the property that if $(x,y)\in R$ and $(x,y')\in R$, then $y=y'$.

A class function need not be a [[set]].

Class functions are an important concept when formalizing [[category theory]] on [[set theory|set-theoretic]] [[foundations]] without using [[universes]], especially when

\begin{itemize}%
\item constructing functors step by step, e.g. by first constructing an assignment between two classes of objects, and then compatibly augmenting this assignment to make it a functor,


\item most especially when constructing \emph{functors on [[functor categories]]}, whose classes of objects tend to be proper classes.



\end{itemize}
\hypertarget{example_in_category_theory}{}\subsection*{{Example in Category Theory}}\label{example_in_category_theory}

An example is Mac Lane's discussion \hyperlink{MacLane1998}{p. 23} of the category of all [[classes]] in his chapter on [[Foundations]]. There, the class of [[arrow]]s is defined by writing ``its arrows are all functions $f\colon C\rightarrow C'$ between classes''.

\hypertarget{example_in_set_theory}{}\subsection*{{Example in Set Theory}}\label{example_in_set_theory}

Arguably the prototypical example of an essential use of the concept of class functions in [[set theory]] is [[Easton's theorem]] \hyperlink{Jech2002}{Theorem 15.18}. on the behaviour of the class function $x\mapsto 2^x$ on the class of all cardinals. (In Easton's theorem, the domain of the class function in question is the proper class of all [[regular cardinals]].)

Often, authors resort to synonyms like [[operation]] or [[assignment]] when writing about proper classes, with \hyperlink{MacLane1998}{p. 23} being a counterexample.

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

\begin{itemize}%
\item [[Saunders Mac Lane]]. \emph{[[Categories for the Working Mathematician]]} Second Edition. Springer (1998)


\item T. Jech. Set theory. Third Edition. Springer (2002)



\end{itemize}


\end{document}