Relationship between Category theory and Axiomatic set theory

I've recently started learning Category theory- and I have a pondering- wondering if anyone can help.

Is it possible for two categories to satisfy two different set-axiom system. Namely- is it possible for two distinct categories $SetZFC$ and $SetZF\neg C$ to exists? One satisfying AC and the other negating it?

And finally, and more generally- Are categories subjected to any a-priory set of set-axioms? or are those axioms given as information in a given category?

Thanks

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The question seems to me can you describe the difference in the language of categories? In $Set(ZFC)$ every epimorphism splits, in $Set(ZF\neg C)$ not.

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I think your second is supposed to be $Set(ZF\neg C)$... – Arturo Magidin Mar 23 '11 at 22:38
Thanks, indeed. – scineram Mar 24 '11 at 12:00

Yes, this is modelable in topos theory. The axiom of choice for toposes says that the any infinite product of non-initial objects is non-initial (one way to state it) or that every epi splits (as noted above). Look up Bill Lawvere's ETCS, which has a choicified version and an non-choicified version.

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Yes, this is certainly possible.

Take $\mathrm{ZF}+\mathrm{AD}$ as foundation, where $\mathrm{AD}$ is the axiom of determinacy. Then we can refute the axiom of choice. Now let $\mathbf{Set}$ denote the category of all sets, with functions as morphisms. Then $\mathbf{Set}$ satisfies: "Some epimorphisms fail to split."

We can also define $L$, Goedel's constructible universe, and this satisfies $\mathrm{ZFC}$ (in fact, it satisfies $\mathrm{ZF}+(V=L).$ See also, axiom of constructibility). So write $\mathbf{Set}^L$ for the category of all constructible sets with constructible functions as morphisms. Then $\mathbf{Set}^L$ satisfies the statement "all epimorphisms split."

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