By contrast, a projective object is an object such that AC holds when making choices indexed by .
Confusingly, some constructivists use “choice set” to mean what we call a “projective set”.
One way to state the axiom of choice is that every entire relation from to (so that every element of is related to some element of ) contains (the graph of) a function . With this terminology, we may say that
Equivalently (at least, in a topos) is choice iff it has a choice function: a function such that for all . Here is the object of all inhabited subsets of . We can also say that an object is choice if and only if it is classically well-orderable (that is, it admits a total order where every inhabited subset has a least element); see D4.5.13 in the Elephant.
Peter Freyd, Choice and Well-ordering , APAL 35 (1987) pp.149-166.
M.-M. Mawanda, Well-ordering and Choice in Toposes , JPAA 50 (1988) pp.171-184.