Contents

foundations

# Contents

## Idea

The assumption that every set has a 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 way).

## 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 $Q$.

## Relationships to other axioms

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

• 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 $(Set/X)_{surj}$.

• A ΠW-pretopos satisfying WISC is a predicative topos.

• Since Michael Rathjen proves that SVC implies AMC (at least in ZF), SVC therefore also implies WISC.

• WISC also follows from the assertion that the free exact completion of $Set$ is 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.”

• WISC implies that the category of anafunctors between any two small categories is essentially small; see here, or below.

• 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 (Karagila).

### Applications

#### Local smallness of anafunctor categories

###### Proposition

WISC implies the local essential smallness of $Cat_ana$, the bicategory of categories and anafunctors.

###### Proof

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

$(j,f) : X \stackrel{j}{\leftarrow} X[U] \stackrel{f}{\to} Y$

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

$\array{ X[V] & \to & X[U] \\ & k \searrow & \downarrow j\\ && X }$

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.

#### Existence of higher inductive types

Swan showed that WISC implies the existence of W-types with reductions?, a kind of simple higher inductive type.

## 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

• For all objects $a$ of $C$, $(C/a)_{cov}$ has a weakly initial set.

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

###### Example

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.)

And now a non-example

###### Example

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 (Stacks Project Tag 0BBK).

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.

## In other categories - internal version

To consider an 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 van den Berg and Moerdijk, a 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 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.

The following two papers give models of set theory (without large cardinals) in which WISC fails.

• Asaf Karagila, Embedding Orders Into Cardinals With $DC_\kappa$, Fund. Math. 226 (2014), 143-156, doi:10.4064/fm226-2-4, arXiv:1212.4396.

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