# Which presheaf toposes satisfy the axiom of choice?

Which toposes of presheaves $$\mathbf{Set}^{C^\mathrm{op}}$$ satisfy the axiom of choice (every epimorphism splits)?

Can one formulate a condition on $$C$$ that yields a necessary or sufficient condition for $$\mathbf{Set}^{C^\mathrm{op}}$$ satisfying the axiom of choice?

In $$\mathbf{Sets}^{\mathcal{C}^\mathrm{op}}$$ every epimorphism splits if and only if $$\mathcal{C}$$ is discrete (or equivalent to a discrete category).
To see this, you can apply the more general result that in a Grothendieck topos $$\mathcal{E}$$ every epimorphism splits if and only if $$\mathcal{E}$$ is equivalent to a topos of sheaves on a complete Boolean algebra, see Theorem 2.2 here. In the literature, this is called the external axiom of choice (there is also an internal one).
An example of a complete Boolean algebra is the power set $$\mathcal{P}(S)$$ of some set $$S$$. The topos of sheaves on $$\mathcal{P}(S)$$ is equivalent to the topos of sheaves on the discrete topological space $$S$$. This topos is also equivalent to a presheaf topos, namely to $$\mathbf{Sets}^{S^\mathrm{op}}$$ where $$S$$ is interpreted as a discrete category.
The above example is the only one. The reason is that presheaf toposes have enough points, and the power sets $$\mathcal{P}(S)$$ are the only complete Boolean algebras such that the topos of sheaves on it has enough points.
So if $$\mathbf{Sets}^{\mathcal{C}^\mathrm{op}}$$ satisfies the external axiom of choice, then $$\mathbf{Sets}^{\mathcal{C}^\mathrm{op}}\!\simeq \mathbf{Sets}^{\mathcal{S}^\mathrm{op}}$$ where $$S$$ is seen as a discrete category. From this it follows that $$\mathcal{C}\simeq S$$, because a category is determined by the topos of presheaves on it, up to idempotent completion.