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.
The “global version” of the axiom 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.
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 ordinals. Ordinary AC is all that is needed to ensure that every set is bijective to a von Neumann ordinal, and that ordinal can be uniquely determined as the smallest in its cardinality class, but in order to additionally choose for every set a bijection to its cardinality, we need global AC.
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).
Mike Shulman: Is there a “local” version of SCSA which follows from non-global AC and non-global SVC? Is there a structural version?