Page "Axiom of choice" Paragraph 71

There are several results in category theory which invoke the axiom of choice for their proof.

These results might be weaker than, equivalent to, or stronger than the axiom of choice, depending on the strength of the technical foundations.

For example, if one defines categories in terms of sets, that is, as sets of objects and morphisms ( usually called a small category ), or even locally small categories, whose hom-objects are sets, then there is no category of all sets, and so it is difficult for a category-theoretic formulation to apply to all sets.

On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.