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\begin{document}

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\section*{axiom of multiple choice}

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

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

\hypertarget{the_axiom_of_multiple_choice}{}\section*{{The axiom of multiple choice}}\label{the_axiom_of_multiple_choice}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{relationships_to_other_axioms}{Relationships to other axioms}\dotfill \pageref*{relationships_to_other_axioms} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{axiom of multiple choice} (AMC) is a weak version of the [[axiom of choice]] which can hold in [[constructive mathematics]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

A set-indexed family $\{D_c\}_{c\in C}$ of sets is said to be a \emph{collection family} if for any $c\in C$ and any [[surjection]] $E\twoheadrightarrow D_c$, there exists a $c'\in C$ and a surjection $D_{c'}\twoheadrightarrow D_c$ which factors through $E$.

Depending on the author, the \emph{axiom of multiple choice} is one of the following statements:

\begin{enumerate}%
\item for every set $X$, there exists a collection family $\{D_c\}_{c\in C}$ such that $X\cong D_c$ for some $c$ ([[Michael Rathjen]]`s formulation, attributed to [[Peter Aczel]] and [[Alex Simpson]]), or


\item for every set $X$, there exists a collection family $\{D_c\}_{c\in C}$, with $C$ inhabited, and a family of surjections $\{D_c \to X\}_{c\in C}$ (the formulation originally given by [[Ieke Moerdijk]] and [[Erik Palmgren]]), or


\item for every set $X$, the full subcategory $(Set/X)_{surj}$ of the slice category $Set/X$ consisting of the surjections has a weakly initial set (in [[Benno van den Berg]]`s formulation; this is also called [[WISC]]).



\end{enumerate}
The nLab uses the initialization AMC to cover either the first two formulations.

[[Mike Shulman]]: Are the first two the same? If not, why are they given the same name?

[[Peter LeFanu Lumsdaine]]: Yes, they are equivalent. For any $X$, given a collection family $\{D_c\}_{c \in C}$ including $X$, then the family $\{D_c\}_{(c \in C, f : D_c \twoheadrightarrow X)}$” is an inhabited collection family equipped with surjections to $X$. Conversely, given an inhabited collection family equipped with surjections to $X$, throwing $X$ into the family gives a collection family including $X$.

The third is a weaker condition, and while some may refer to as a ``weak axiom of multiple choice'', van den Berg obviously does not; he calls his the AMC and the Moerdijk-Palmgren formulation rather the ``strong axiom of multiple choice''.

\hypertarget{relationships_to_other_axioms}{}\subsection*{{Relationships to other axioms}}\label{relationships_to_other_axioms}

\begin{itemize}%
\item Note that $P$ is a [[projective set]] if and only if the singleton family $\{P\}$ is a collection family. Therefore, since AC is equivalent to ``all sets are projective,'' it implies AMC.


\item An extension of this argument shows that [[COSHEP]] is sufficient to imply AMC.


\item The [[Reflection Principle]] (RP) is equivalent to AMC (the one called strong AMC by van den Berg). RP is motivated by the regular extension principle (REA) from constructive set theory. RP states that every map belongs to a representable class of small maps.


\item However, AMC does not imply [[countable choice]] or any of the other usual consequences of AC.


\item Rathjen proves that [[SVC]] also implies AMC. It follows that AMC holds in ``most'' models of set theory.


\item AMC implies [[WISC]], and therefore also implies that the category of [[anafunctors]] between two [[small categories]] is [[essentially small category|essentially small]]. Thus WISC may be termed ``weak axiom of multiple choice''.


\item A [[ΠW-pretopos]] satisfying the (weak) axiom of multiple choice is a \emph{[[predicative topos]]}, or removing the word ``weak'', we may speak of a strong predicative topos.



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

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Erik Palmgren]], \emph{Type theories, toposes and constructive set theory: predicative aspects of AST} (2000) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.8934}{web})


\item Rathjen, ``Choice principles in constructive and classical set theories''



\end{itemize}
In

\begin{itemize}%
\item [[Benno van den Berg]], \emph{Predicative toposes} (\href{http://arxiv.org/abs/1207.0959}{arXiv:1207.0959})

\end{itemize}
[[WISC]] is called the ``axiom of multiple choice''.

category: foundational axiom

[[!redirects axiom of multiple choice]] [[!redirects multiple choice]] [[!redirects AMC]]



\end{document}