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In the paper Quantifiers and Sheaves by Lawvere, at the bottom of the second page, the author writes:

"... the condition that every epi splits, which geometrically we would call 0-dimensionality and logically we would call the axiom of choice."

What is meant by this geometric interpretation of choice as zero-dimensionality? I'm guessing it has something to do with fibers but can't figure it out myself.

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Here are two possible motivating examples.

First, for any topos $\mathcal{E}$, if all epimorphisms in $\mathcal{E}$ split, then $\mathcal{E}$ is a boolean topos. In particular, for a topological space $X$, if all epimorphisms in $\mathbf{Sh} (X)$ split, then every open subset of $X$ is also closed (and vice versa), so $X$ is a disjoint union of indiscrete spaces. The converse is straightforward. Thus, all epimorphisms in $\mathbf{Sh} (X)$ split if and only if $X$ is a disjoint union of indiscrete spaces.

Second, for a commutative ring $A$, if all epimorphisms in $\mathbf{Mod} (A)$ split, then every closed subset of $\operatorname{Spec} A$ is also open (and vice versa), so the Krull dimension of $A$ is zero, therefore $\operatorname{Spec} A$ is finite and discrete, and hence $A$ is a direct product of finitely many fields. Conversely, if $A$ is a direct product of finitely many fields then all epimorphisms in $\mathbf{Mod} (A)$ split.

I suppose the point is that if the category of representations of some geometric object satisfies the axiom of choice, then that object cannot have a very complicated internal structure.

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  • $\begingroup$ I don't see why Spec(A) should be finite and discrete, unless A is noetherian. And how to deduce that A is a finite product of fields, unless A is reduced. $\endgroup$ – HeinrichD Sep 23 '16 at 11:30

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