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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?

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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.

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    $\begingroup$ There is also the result that a presheaf topos is boolean if and only if it is presheaves on a groupoid. $\endgroup$
    – Zhen Lin
    Dec 30, 2021 at 0:31

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