# nLab small cardinality selection axiom

The small cardinality selection axiom

# The small cardinality selection axiom

## Idea

The small cardinality selection axiom (SCSA) is a weak form of the axiom of choice, which asserts in a certain precise way that choice fails “in at most a small way”. It was introduced by Michael Makkai for the study of anafunctors, and thus it also has consequences for the existence of stack completions.

## Definition

The “global version” of SCSA states:

Therefore, we have a class function assigning to each set $A$, another set ${\Vert A\Vert}$ and a bijection $A \cong {\Vert A \Vert}$, in such a way that ${\Vert{-}\Vert}$ takes only set-many values on each isomorphism class in Set.

There is also a “local version” of SCSA which can be stated as follows. Given a function $f:A\to I$, say that a function $g:B\to J$ equipped with $\phi:J\to I$ is a local pullback of $f$ if there is a pair of pullback squares

in which $p$ is surjective. Then the local SCSA states:

• For any $f:A\to I$, the category whose objects are local pullbacks of $f$ and whose morphisms are pullback squares over $I$ has a weakly terminal object.

In more set-theoretic language, given a family of sets $(A_i)_{i\in I}$, define a family of sets $(B_j)_{j\in J}$ equipped with $\phi:J\to I$ to be a local reindexing of $(A_i)$ if for each $j\in J$ there exists an isomorphism $B_j \cong A_{\phi(j)}$ (but there is not necessarily a function picking out such an isomorphism). Then the local SCSA states that for any $(A_i)_{i\in I}$ has a local reindexing $(B_j)_{j\in J}$ with $\phi:J\to I$ such that for any other local reindexing $(C_k)_{k\in K}$ with $\psi:K\to I$, there exists a function $\chi : K\to J$ such that $\phi\chi = \psi$ and a specified family of isomorphisms $C_k \cong B_{\chi(k)}$.

To deduce the local version from the global version, given $(A_i)_{i\in I}$ let $J$ be the set of pairs $(i, Y)$ such that $Y = \Vert X\Vert$ for some $X\cong A_i$, with $\phi$ the first projection and $B_{(i,Y)} = Y$. Then given $(C_k)_{k\in K}$ with $\psi:K\to I$ such that $C_k \cong A_{\psi(k)}$, define $\chi(k) = (\psi(k),\Vert C_k\Vert)$.

## Relationship to other forms of choice

• In the presence of the axiom of global choice in material set theory, the category Set has a skeleton, namely the category of von Neumann cardinals, which of course implies the global SCSA. Similarly, the ordinary axiom of choice implies that every set is bijective to a unique von Neumann cardinal, and we can choose such a bijection uniformly for any small family of sets, which implies the local SCSA.

• Global SCSA also follows from the “global” version of the axiom of small violations of choice, as proven in Makkai’s paper (attributed to the referee). Makkai claims there is a local form of SCSA that follows from the local SVC. I have not attempted to show that this is true for the above local SCSA, although it is a guess at the sort of axiom Makkai might have had in mind.

## Consequences

• SCSA implies that the anabicategory of categories and anafunctors is locally essentially small and cartesian closed. Specifically, local SCSA implies that for any two categories $X,A$ there is a small set of anafunctors from $X$ to $A$ such that any such anafunctor is isomorphic to one in this set. The idea is that we can replace any anafunctor by a saturated one, in which case the set of “specifications that $F(x)=a$” for any $x\in X, a\in A$ is, if nonempty, a torsor over $Iso(a,a)$, and hence isomorphic to it. Thus, applying local SCSA to the family $(Iso(a,a))_{a\in A}$, we obtain a “universal” set of specifications for saturated anafunctors.

This implies that there exists a full subcategory of $Cat_{ana}(X,A)$ whose inclusion is a weak equivalence, and that an exponential $A^X$ exists. If local SCSA holds in “functional” form, meaning there is a class-function specifying weakly terminal local reindexings, then this implies that $Cat_{ana}$ is weakly equivalent (i.e. ana-equivalent) to a locally small bicategory and that it admits a functor $(X,A) \mapsto A^X$. (Of course, global SCSA implies functional local SCSA.)

## Models

Makkai states only the global SCSA, but claims that there is a local version of it that holds in all Grothendieck toposes. The local form of SCSA stated above is a guess at the one Makkai might have had in mind. I have not attempted to verify that it is true in all Grothendieck toposes.