basic constructions:
strong axioms
further
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).
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 $Q$.
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).
WISC implies the local essential smallness of $Cat_ana$.
Let $X,Y$ be small categories and consider the category $Cat_{ana}(X,Y)$, with objects which are spans
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
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.
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
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.
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
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.
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).
Benno van den Berg, Ieke Moerdijk, The Axiom of Multiple Choice and Models for Constructive Set Theory, arXiv. In this paper WISC is called the “axiom of multiple choice”.
Thomas Streicher, Realizability Models for $CZF + \neg Pow$, unpublished note. In this note WISC is called $TTCA_f$ ($TTCA$ stands for “type-theoretic collection axiom).
Benno van den Berg, WISC is independent from ZF, PDF
In
WISC is called the “axiom of multiple choice”.
The following two papers give models of set theory (without large cardinals) in which WISC fails
The Stacks Project shows how to construct a counterexample to WISC from any set of fpqc covers of an affine scheme.