# Is Choice an assumption or determined by category?

Is the axiom of choice an assumption, that one may "freely" choose (eg, ZFC) or discard (eg, ZF, ZF+AD), or is it determined by the nature of the categories being considered?

The latter view is expoused in Lawevere & Rosebrugh's Sets for Mathematics where it's stated that Choice is false in categories with "internal motion and cohesion", as opposed to, eg, the category of constant sets where Choice is true.

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Choice is a property of a category, and if you so wish, the universe is also a category, so it is also a property of the universe. –  Zhen Lin Dec 30 '12 at 5:27
An instance in which the Axiom of Choice seems too valuable to discard is the category of left $R$-modules. By Zorn's Lemma, we have a useful criterion for testing whether or not a given left $R$-module is injective. It is called Baer's Criterion, and it states that a left $R$-module $M$ is injective if and only if for any left ideal $I$ of $R$, every $R$-homomorphism $h: I \to M$ can be extended to an $R$-homomorphism $\tilde{h}: R \to M$. This is such a useful and powerful result that the category seems to demand for it, and hence, for some form of AC. –  Haskell Curry Dec 30 '12 at 7:30
@ZhenLin, in Goldblatt's Topoi and Awodey's Category Theory it is mentioned - without resolution - that the concept of a category of categories gets to a logical cliff similar to Russell's paradox in set theory. I believe Woodin has also written that the universe of all sets is "fiction". What do you mean by "the universe is also a category"? –  alancalvitti Dec 30 '12 at 17:51

Whether or not choice holds may affect properties a particular category may have. For a simple example, in the category $Set$ of sets and functions every epimorphism admits a section if, and only if, the axiom of choice holds.
@Ittay, so does Choice hold in $Set$ or not? Is it determined by category axioms? My amateur reading of Lawvere's material is that choice is true in $Set$ but not in categories of variable sets. –  alancalvitti Dec 30 '12 at 19:20