nLab
WISC

Context

Foundations

Idea
Statement (for Sets)
Relationships to other axioms
Relation to local smallness of anafunctor categories
In other sites - external version
In other categories - internal version
References

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 ${f_{i} : P_{i} ↠ X}_{i \in I}$ such that for any surjection $Q ↠ X$ , there exists some $f_{i}$ which factors through $Q$ .

Relationships to other axioms

WISC is implied by COSHEP, since any surjection $P ↠ X$ 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 $D_{c} ↠ X$ is weakly initial in $(Set / X)_{surj}$ .
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 $Set$ 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.

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 ${Cat}_{ana}$ .

Proof

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

(j, f) : X \overset{j}{\leftarrow} X [U] \overset{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{matrix} X [V] & \to & X [U] \\ k ↘ & ↓ j \\ X \end{matrix}

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.

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 $p : Y \to X$ such that for any $x \in X$ there exist an open set $U ∋ x$ 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 $∐_{U \in 𝒰} U \to X$ 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 $U_{x} ∋ x$ over which $p$ has a section, resulting in an open cover $𝒰 = {U_{x} ∣ x \in X}$ of $X$ for which $∐_{U \in 𝒰} U \to X$ factors through $p$ . (If $Set$ 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 $Ana (C, J)$ 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).

References

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

Benno van den Berg, Predicative toposes (arXiv:1207.0959)

WISC is called the “axiom of multiple choice”.

Asaf Karagila, Embedding Orders Into Cardinals With ${DC}_{κ}$ , arXiv:1212.4396.

category: foundational axiom

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

nLab
WISC

Context

Foundations

The basis of it all

Foundational axioms

Removing axioms

Contents

Idea

Statement (for Sets)

Relationships to other axioms

Relation to local smallness of anafunctor categories

Proposition

Proof

In other sites - external version

Example

In other categories - internal version

References

nLab WISC

Context

Foundations

The basis of it all

Foundational axioms

Removing axioms

Contents

Idea

Statement (for Sets)

Relationships to other axioms

Relation to local smallness of anafunctor categories

Proposition

Proof

In other sites - external version

Example

In other categories - internal version

References

nLab
WISC