nLab 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:

• The (large) category Set of sets is strongly equivalent to a full subcategory of itself, each of whose isomorphism classes contains only a set of objects.

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.

References

• Michael Makkai, Avoiding the axiom of choice in general category theory, Journal of Pure and Applied Algebra 108 isse 2 (1996) pp 109-173, doi:10.1016/0022-4049(95)00029-1, (author’s page)

category: foundational axiom