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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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    @ZhenLin, in Goldblatt's *Topoi* and Awodey's *Category Theory* it is $m$entioned - 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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    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