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\begin{document}

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\section*{WISC}

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

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

[[!include foundations - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement_for_sets}{Statement (for Sets)}\dotfill \pageref*{statement_for_sets} \linebreak
\noindent\hyperlink{relationships_to_other_axioms}{Relationships to other axioms}\dotfill \pageref*{relationships_to_other_axioms} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{local_smallness_of_anafunctor_categories}{Local smallness of anafunctor categories}\dotfill \pageref*{local_smallness_of_anafunctor_categories} \linebreak
\noindent\hyperlink{existence_of_higher_inductive_types}{Existence of higher inductive types}\dotfill \pageref*{existence_of_higher_inductive_types} \linebreak
\noindent\hyperlink{in_other_sites__external_version}{In other sites - external version}\dotfill \pageref*{in_other_sites__external_version} \linebreak
\noindent\hyperlink{in_other_categories__internal_version}{In other categories - internal version}\dotfill \pageref*{in_other_categories__internal_version} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The assumption that every [[set]] has a \textbf{Weakly Initial Set of Covers}, or $WISC$, is a weak form of the [[axiom of choice]]. Like the [[axiom of multiple choice]] and the axiom of [[small violations of choice]] (which both imply it), it says intuitively that ``$AC$ fails to hold only in a small way'' (i.e. not in a [[proper class|proper-class]] way).

\hypertarget{statement_for_sets}{}\subsection*{{Statement (for Sets)}}\label{statement_for_sets}

Precisely, $WISC$ is the statement that for any [[set]] $X$, the [[full subcategory]] $(Set/X)_{surj}$ of the [[slice category]] $Set/X$ consisting of the [[surjections]] has a [[weakly initial set]]. In other words, there is a family of surjections $\{f_i\colon P_i \twoheadrightarrow X\}_{i\in I}$ such that for any surjection $Q\twoheadrightarrow X$, there exists some $f_i$ which factors through $Q$.

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

\begin{itemize}%
\item WISC is implied by [[COSHEP]], since any surjection $P\twoheadrightarrow X$ such that $P$ is projective is necessarily a weakly initial (singleton) set in $(Set/X)_{surj}$.


\item WISC is also implied by the [[axiom of multiple choice]] (which is in turn implied by COSHEP). For if $X$ is in some collection family $\{D_c\}_{c\in C}$, then the family of all surjections of the form $D_c \twoheadrightarrow X$ is weakly initial in $(Set/X)_{surj}$.


\item A [[ΠW-pretopos]] satisfying WISC is a \emph{[[predicative topos]]}.


\item Since [[Michael Rathjen]] proves that [[SVC]] implies [[AMC]] (at least in [[ZF]]), SVC therefore also implies WISC.


\item WISC also follows from the assertion that the [[free exact completion]] of $Set$ is [[well-powered category|well-powered]], which in turn follows from assertion that $Set$ has a [[generic proof]] (so that $Set_{ex/lex}$ is a [[topos]]). Both of these can also be regarded as saying that choice is only violated ``in a small way.''


\item WISC implies that the category of [[anafunctors]] between any two [[small categories]] is [[essentially small category|essentially small]]; see \href{/nlab/show/anafunctor#SizeQuestions}{here}, or below.


\item WISC implies (in [[ZF]]) that there exist arbitrarily large [[regular cardinals]]. Therefore, WISC is not provable in ZF, as Moti Gitik constructed a model of ZF with only one regular cardinal, using large cardinal assumptions. A proof without large cardinals was given in (\hyperlink{Karagila14}{Karagila}).



\end{itemize}
\hypertarget{applications}{}\subsubsection*{{Applications}}\label{applications}

\hypertarget{local_smallness_of_anafunctor_categories}{}\paragraph*{{Local smallness of anafunctor categories}}\label{local_smallness_of_anafunctor_categories}

\begin{prop}
\label{}\hypertarget{}{}
WISC implies the local essential smallness of $Cat_ana$, the bicategory of categories and [[anafunctors]].

\end{prop}
\begin{proof}
Let $X,Y$ be small categories and consider the category $Cat_{ana}(X,Y)$, with objects which are spans

\begin{displaymath}
(j,f) : X \stackrel{j}{\leftarrow} X[U] \stackrel{f}{\to} Y
\end{displaymath}
where $X[U] \to X$ is a surjective-on-objects, [[fully faithful]] functor. The underlying map on object sets is $U \to X_0$. By WISC there is a surjection $V \to X_0$ and a map $V\to U$ over $X_0$. We can thus define a commuting triangle of functors

\begin{displaymath}
\itexarray{ 
    X[V] &  \to  &  X[U]    \\ 
    & k \searrow & \downarrow j\\ 
    && X 
  }
\end{displaymath}
where $X[V] \to X$ is the canonical fully faithful functor arising from $V\to X_0$ (the arrows of $X[V]$ are given by $V^2 \times_{X_0^2} X_1$). This gives rise to a transformation from $(j,f)$ to a span with left leg $k$. Thus $Cat_{ana}(X,Y)$ is equivalent to the [[full subcategory]] of anafunctors where the left leg has as object component an element of the weakly initial set of surjections. Since there is only a set of functors $X[V] \to Y$ for each $V\to X_0$, this subcategory is small.

\end{proof}
\hypertarget{existence_of_higher_inductive_types}{}\paragraph*{{Existence of higher inductive types}}\label{existence_of_higher_inductive_types}

\hyperlink{Swan18}{Swan} showed that WISC implies the existence of [[W-types with reductions]], a kind of simple [[higher inductive type]].

\hypertarget{in_other_sites__external_version}{}\subsection*{{In other sites - external version}}\label{in_other_sites__external_version}

Let $(C,J)$ be a [[site]] with a singleton [[Grothendieck pretopology]] $J$. It makes sense to consider a version of WISC for $(C,J)$, along the lines of the following: Let $(C/a)_{cov}$ be the full subcategory of the slice category $C/a$ consisting of the covers. WISC then states that

\begin{itemize}%
\item For all objects $a$ of $C$, $(C/a)_{cov}$ has a weakly initial set.

\end{itemize}
This definition is called \emph{external} because it refers to an external category of sets. This is to be contrasted with the \emph{internal} version of WISC, discussed below.

\begin{example}
\label{}\hypertarget{}{}
Assuming AC for $Set$, the category $Top$ with any of its usual pretopologies satisfies 'internal WISC'. Consider, for instance, the pretopology in which the covers are the maps admitting local sections, i.e. those $p\colon Y\to X$ such that for any $x\in X$ there exist an open set $U\ni x$ such that $p^{-1}(U)\to U$ is split epic. If $Set$ satisfies AC, then a weakly initial set in $Top/_{cov}X$ is given by the set of all maps $\coprod_{U\in \mathcal{U}} U \to X$ where $\mathcal{U}\subset \mathcal{P}(X)$ is an open cover of $X$. For if $p\colon Y\to X$ admits local sections, then for each $x\in X$ we can choose an $U_x \ni x$ over which $p$ has a section, resulting in an open cover $\mathcal{U} = \{U_x \mid x\in X\}$ of $X$ for which $\coprod_{U\in \mathcal{U}} U \to X$ factors through $p$. (If $Set$ merely satisfies WISC itself, then a more involved argument is required.)

\end{example}
And now a non-example

\begin{example}
\label{}\hypertarget{}{}
The category of [[affine schemes]] can be equipped with the [[fpqc topology]] (so this is the [[fpqc site]] over $Spec(\mathbb{Z})$). This does not satisfy WISC. Namely, given any set of fpqc covers of $Spec(R)$, there is a surjective fpqc map which is refined by none of the given covering families (\hyperlink{Stacks0BBK}{Stacks Project Tag 0BBK}).

\end{example}
More generally, for a non-singleton pretopology on $C$, we can reformulate WISC along the lines of 'there is a set of covering families weakly initial in the category of all covering families of any object'.

Given a site $(C,J)$ with $J$ subcanonical, and $C$ finitely complete, we can define a (weak) 2-category $Ana(C,J)$ of internal categories, anafunctors and transformations. If WISC holds for $(C,J)$, then $Ana(C,J)$ is locally essentially small.

\hypertarget{in_other_categories__internal_version}{}\subsection*{{In other categories - internal version}}\label{in_other_categories__internal_version}

To consider an \emph{internal} version of WISC, which doesn't refer to an external notion of set, one needs to assume that the ambient category $C$ has a strong enough [[internal logic]], such as a pretopos (this is the context in which van den Berg and Moerdijk work). Then the ordinary statement of WISC in set can be written in the internal logic, using the [[stack semantics]], as a statement about the objects and arrows of $C$. It is in this form that WISC is useful as a replacement choice principle in intuitionistic, constructive or predicative set theory, as these are modelled on various topos-like categories (or in the case of \hyperlink{vdBM12}{van den Berg and Moerdijk}, a [[algebraic set theory|category of classes]], although this is not necessary for the approach).

Importantly, the internal form of WISC is stable under many more categorical constructions than other forms of choice. For instance, any [[Grothendieck topos]] or [[realizability topos]] inherits WISC from its base topos of sets (even if the latter is constructive or even predicative); see \hyperlink{vdBerg12}{van den Berg}. This is in contrast to nearly all other choice principles, including weaker forms such as [[COSHEP]] and [[SVC]], which fail in at least some Grothendieck toposes even when the base is a model of ZFC.

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

\begin{itemize}%
\item [[Benno van den Berg]], [[Ieke Moerdijk]], \emph{The Axiom of Multiple Choice and Models for Constructive Set Theory}, \href{http://arxiv.org/abs/1204.4045}{arXiv}. In this paper WISC is called the ``axiom of multiple choice''.


\item [[Thomas Streicher]], \emph{Realizability Models for $CZF + \neg Pow$}, \href{http://www.mathematik.tu-darmstadt.de/~streicher/CIZF/rmczfnp.pdf}{unpublished note}. In this note WISC is called $TTCA_f$ ($TTCA$ stands for ``type-theoretic collection axiom).


\item [[Benno van den Berg]], \emph{WISC is independent from ZF}, \href{http://www.staff.science.uu.nl/~berg0002/papers/WISC.pdf}{PDF}


\item [[Benno van den Berg]], \emph{Predicative toposes} (\href{http://arxiv.org/abs/1207.0959}{arXiv:1207.0959}). In this paper WISC is called the ``axiom of multiple choice''.


\item [[Andrew Swan]], \emph{W-Types with Reductions and the Small Object Argument}, 2018, \href{https://arxiv.org/abs/1802.07588}{arxiv:1802.07588}



\end{itemize}
The following two papers give models of set theory (without large cardinals) in which WISC fails.

\begin{itemize}%
\item Asaf Karagila, \emph{Embedding Orders Into Cardinals With $DC_\kappa$}, Fund. Math. 226 (2014), 143-156, doi:\href{http://dx.doi.org/10.4064/fm226-2-4}{10.4064/fm226-2-4}, \href{http://arxiv.org/abs/1212.4396}{arXiv:1212.4396}.

\end{itemize}
\begin{itemize}%
\item [[David Roberts]], \emph{The weak choice principle WISC may fail in the category of sets}, \href{http://link.springer.com/journal/11225}{Studia Logica} Volume 103 (2015) Issue 5, pp 1005-1017, doi:\href{http://dx.doi.org/10.1007/s11225-015-9603-6}{10.1007/s11225-015-9603-6} \href{http://arxiv.org/abs/1311.3074}{arXiv:1311.3074}

\end{itemize}
The Stacks Project shows how to construct a counterexample to WISC from any set of fpqc covers of an affine scheme.

\begin{itemize}%
\item [[The Stacks Project]], \href{http://stacks.math.columbia.edu/tag/0BBK}{Tag 0BBK}

\end{itemize}
category: foundational axiom

[[!redirects WISC]] [[!redirects WISCs]] [[!redirects weakly initial set of covers]] [[!redirects weakly initial sets of covers]] [[!redirects axiom of weakly initial sets of covers]]

[[!redirects SOSHWIS]]



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