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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.2012-12-30
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    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.2012-12-30
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    @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"?2012-12-30

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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.

Also whether or not one can do certain constructions depends on whether or not choice is available. For instance, collecting all universal solutions for a given functor into a single left adjoint requires (often a very strong variant of) the axiom of choice.

There is also a question of chicken and egg: what comes first, the category one studies or the objects and arrows in it. Looking at it this way choice might be dictated by either wanting the category to have certain properties or by wanting the objects to have certain properties.

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    I totally agree with your chicken-or-egg analogy. If we need a particular result to hold in a category because it is useful and makes life much simpler, and if some variant of AC is required for it to be true, then it is often prudent to invoke AC or its aforementioned variant.2012-12-30
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    @HaskellCurry, but typically there are tradeoffs, for example, in ZFC, paradoxical Banach-Tarski decompositions are true, so there's no realistic 3-dimensional geometry. On the other hand, in ZF+AD, there's no nontrivial ultrafilters on the natural numbers, and no real vector space over the rationals (all examples from Herrlich's *Axiom of Choice*). Assumptions have consequences that make one branch simpler and another harder.2012-12-30
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    @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.2012-12-30
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    In the context of topos theory: the category Set of sets (given a particular model of ZF), viewed as a topos, satisfies AC (in the sense that every epimorphism has a section) iff AC holds in the model of ZF.2012-12-30
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    @alancalvitti Or, to put it more bluntly, there is more than one category of sets.2012-12-31
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    @IttayWeiss, what's the correct interpretation of the statement of Lawvere regarding Choice being false in categories of variable sets and categories with internal cohesion. Is this also determined by underlying models as opposed to categorical axioms?2012-12-31
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    @ZhenLin, why are these multiple categories of sets not generally distinguished? Even notationally, there's just $Set$ as opposed to $Set_{ZF}$ vs. $Set_{ZFC}$ vs. $Set_{ZF+AD}$ etc.2012-12-31
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    Because it usually doesn't matter. (Typically $\textbf{Set}$ is assumed to be at the very least a model of ETCS if not the whole of ZFC.)2012-12-31
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    @ZhenLin, you're more familiar w/ the disasters w/ and w/o choice in Herrich's *AC* than I am. Outside of $\textbf {FinSet}$, to the degree that sets are used as "substance" on which to overlay many other fundamental structures, I would say the distinctions do matter, eg, there's no 3-dimensional geometry with $AC$. How does that not matter?2013-01-01
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    In case it's not clear to you, $\textbf{Set}$ denotes _the_ category of sets in which ordinary mathematics happens, and in particular choice is assumed (often in very powerful forms). But if you have different background assumptions that $\textbf{Set}$ can be different. In other words, $\textbf{Set}$ is not an _object_ of study but a _tool_ of study. We may choose from time to time to study models of set theory by category-theoretic means, but these would not denoted by $\textbf{Set}$.2013-01-01