nLab
WISC

Contents

Idea

The assumption that every set has a Weakly Initial Set of Covers, or WISCWISC, 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 ”ACAC fails to hold only in a small way” (i.e. not in a proper-class way).

Statement (for Sets)

Precisely, WISCWISC is the statement that for any set XX, the full subcategory (Set/X) surj(Set/X)_{surj} of the slice category Set/XSet/X consisting of the surjections has a weakly initial set. In other words, there is a family of surjections {f i:P iX} iI\{f_i\colon P_i \twoheadrightarrow X\}_{i\in I} such that for any surjection QXQ\twoheadrightarrow X, there exists some f if_i which factors through QQ.

Relationships to other axioms

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

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

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

  • 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 SetSet is well-powered, which in turn follows from assertion that SetSet has a generic proof (so that Set ex/lexSet_{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.

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 Cat anaCat_ana.

Proof

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

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

where X[U]XX[U] \to X is a surjective-on-objects, fully faithful functor. The underlying map on object sets is UX 0U \to X_0. By WISC there is a surjection VX 0V \to X_0 and a map VUV\to U over X 0X_0. We can thus define a commuting triangle of functors

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

where X[V]XX[V] \to X is the canonical fully faithful functor arising from VX 0V\to X_0 (the arrows of X[V]X[V] are given by V 2× X 0 2X 1V^2 \times_{X_0^2} X_1). This gives rise to a transformation from (j,f)(j,f) to a span with left leg kk. Thus Cat ana(X,Y)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]YX[V] \to Y for each VX 0V\to X_0, this subcategory is small.

In other sites - external version

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

  • For all objects aa of CC, (C/a) cov(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 SetSet, the category TopTop 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:YXp\colon Y\to X such that for any xXx\in X there exist an open set UxU\ni x such that p 1(U)Up^{-1}(U)\to U is split epic. If SetSet satisfies AC, then a weakly initial set in Top/ covXTop/_{cov}X is given by the set of all maps U𝒰UX\coprod_{U\in \mathcal{U}} U \to X where 𝒰𝒫(X)\mathcal{U}\subset \mathcal{P}(X) is an open cover of XX. For if p:YXp\colon Y\to X admits local sections, then for each xXx\in X we can choose an U xxU_x \ni x over which pp has a section, resulting in an open cover 𝒰={U xxX}\mathcal{U} = \{U_x \mid x\in X\} of XX for which U𝒰UX\coprod_{U\in \mathcal{U}} U \to X factors through pp. (If SetSet merely satisfies WISC itself, then a more involved argument is required.)

More generally, for a non-singleton pretopology on CC, 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)(C,J) with JJ subcanonical, and CC finitely complete, we can define a (weak) 2-category Ana(C,J)Ana(C,J) of internal categories, anafunctors and transformations. If WISC holds for (C,J)(C,J), then Ana(C,J)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 CC 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 CC. 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).

References

In

WISC is called the “axiom of multiple choice”.

  • Asaf Karagila, Embedding Orders Into Cardinals With DC κDC_\kappa, arXiv:1212.4396.

Revised on January 2, 2013 11:19:45 by David Roberts (192.43.227.18)