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.
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.
