Editing Axiom of choice (section)

===Category theory===
Several results in [[category theory]] 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]]), then there is no [[category of sets|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.

Examples of category-theoretic statements which require choice include:
*Every small [[category (mathematics)|category]] has a [[skeleton (category theory)|skeleton]].
*If two small categories are weakly equivalent, then they are [[equivalence of categories|equivalent]].
*Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a [[adjoint functors|left-adjoint]] (the Freyd adjoint functor theorem).