What are (if any) equivalent forms of AC (The Axiom of Choice) in Category Theory ? 



Category and the axiom of choice 
Here's a somewhat trivial one, but it is one that category theorists use all the time:



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> Let us say that a functor $F : \mathcal{C} \to \mathcal{D}$ is a **weak equivalence** if it is fully faithful and essentially surjective on objects, and that it is a **strong equivalence** if there exists a functor $G : \mathcal{D} \to \mathcal{C}$ such that $G F \cong \textrm{id}\_{\mathcal{C}}$ and $F G \cong \textrm{id}\_\mathcal{D}$.
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> **Proposition.** In Zermelo set theory with only bounded separation, the following are equivalent:
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> * Every surjection of sets splits.
> * Any weak equivalence between two small categories is a strong equivalence.
> * Any weak equivalence between two small groupoids is a strong equivalence.
> * Any weak equivalence between two small preorders is a strong equivalence.
> * Any weak equivalence between two small setoids is a strong equivalence.
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> Here, by "small" I mean something internal to the set-theoretic universe in question.
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On the other hand, if you're asking for category-theoretic formulations of the axiom of choice *inside* some category of "sets", then there are several:


* The usual formulation just says that every epimorphism in $\textbf{Set}$ splits. This generalises easily to any category.
* In any topos $\mathcal{E}$, one can formulate the axiom schema "every surjection $X \to Y$ splits" in the internal language of $\mathcal{E}$, and this axiom schema is valid if and only if every object is *internally* projective, in the sense that the functor $(-)^X : \mathcal{E} \to \mathcal{E}$ preserves epimorphisms. This is called the **internal axiom of choice**.


The internal axiom of choice holds in $\textbf{Set}$ precisely if the usual axiom of choice holds; this is because $\textbf{Set}$ is a well-pointed topos; but in general the internal axiom of choice is weaker. For example, for any discrete group $G$, the category $\mathbf{B} G$ of all $G$-sets and $G$-equivariant maps is a topos in which the internal axiom of choice holds, but if $G$ is any non-trivial group whatsoever, then there exist epimorphisms in $\mathbf{B} G$ that do not split. (For example, $G \to 1$, where $G$ acts on itself by translation.)


