nLab
WISC

The assumption that every set has a Weakly Initial Set of Covers, or $\mathrm{WISC}$, 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 ”$\mathrm{AC}$ fails to hold only in a small way” (i.e. not in a proper-class way).

Statement (for Sets)

Precisely, $\mathrm{WISC}$ is the statement that for any set $X$, the full subcategory $(\mathrm{Set}/X{)}_{\mathrm{surj}}$ of the slice category $\mathrm{Set}/X$ consisting of the surjections has a weakly initial set. In other words, there is a family of surjections $\{f_i:P_i\twoheadrightarrow X{\}}_{i\in I}$ 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 $(\mathrm{Set}/X{)}_{\mathrm{surj}}$.

• 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 $(\mathrm{Set}/X{)}_{\mathrm{surj}}$.

• Since 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 $\mathrm{Set}$ is well-powered, which in turn follows from assertion that $\mathrm{Set}$ has a generic proof (so that ${\mathrm{Set}}_{\mathrm{ex}/\mathrm{lex}}$ 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.

Relation to local smallness of anafunctor categories

David Roberts: Some of the terms may need to be considered in terms of the bicategory of small categories and anafunctors, rather than the 2-category of small categories and functors, for example ‘essential smallness’.

Proposition

WISC implies the local essential smallness of ${\mathrm{Cat}}_{\mathrm{ana}}$.

Proof

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

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

$$\begin{array}{ccc}X[V]& \to & X[U]\\ & k\searrow & \downarrow j\\ & & X\end{array}$$\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 ${\mathrm{Cat}}_{\mathrm{ana}}(X,Y)$ 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.

In other sites

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 $C{/}_{\mathrm{cov}}a$ be the full subcategory of the slice category $C/a$ consisting of the covers. Internal WISC then states that

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

Example

Assuming AC for $\mathrm{Set}$, the category $\mathrm{Top}$ 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 $p:Y\to X$ 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 $\mathrm{Set}$ satisfies AC, then a weakly initial set in $\mathrm{Top}{/}_{\mathrm{cov}}X$ is given by the set of all maps ${\coprod }_{U\in 𝒰}U\to X$ where $𝒰\subset 𝒫(X)$ is an open cover of $X$. For if $p:Y\to X$ 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 $𝒰=\{U_x\mid x\in X\}$ of $X$ for which ${\coprod }_{U\in 𝒰}U\to X$ factors through $p$. (If $\mathrm{Set}$ merely satisfies WISC itself, then a more involved argument is required.)

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 $\mathrm{Ana}(C,J)$ of internal categories, anafunctors and transformations. If WISC holds for $(C,J)$, then $\mathrm{Ana}(C,J)$ is locally essentially small.

category: foundational axiom