What are (if any) equivalent forms of AC (The Axiom of Choice) in Category Theory ?
[Math] Category and the axiom of choice
axiom-of-choicect.category-theoryset-theory
axiom-of-choicect.category-theoryset-theory
What are (if any) equivalent forms of AC (The Axiom of Choice) in Category Theory ?
Best Answer
Here's a somewhat trivial one, but it is one that category theorists use all the time:
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:
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.)