Internally Inhabited Object Which Is Not Externally Inhabited

Question

What is the simplest example of a topos with an object $X$ which is internally ($X\to \mathbf{1}$ is epi) but not externally (no $\mathbf{1}\to X$) inhabited?

I think that for all categories $C$, in the topos $\text{Set}^C$ the externally and internally inhabited objects are the same, namely the ones with no empty sets in the image of $C_0$.

Then I thought, what if we take $\text{Set}^2$ but allow only maps $f_i:X_i\to Y_i$ where $f_0\cup f_1$ is a map, i.e. same elements in $X_0$ and $X_1$ must be mapped to same elements. This way, there is no map $\mathbf{1}=(1,1)\to(\{a\},\{b\})$ for $a\neq b$ but $(\{a\},\{b\})\to \mathbf{1}$ is epi. The question I couldn't answer, is this category even a topos?

Another idea was to take $\text{Set}^\omega$ and use some negation of countable AC so that there are sequences $X$ of non-empty $X_i$ with no choice function hence no map $\mathbf{1}\to X$, but I don't know how to formalize this.

Answer 1

Consider the integers considered as an additive group considered as a one-object category $\mathsf Z$. Now $\mathrm{Psh}(\mathsf Z)$ is a topos. Its objects are sets $X$ equipped with an arbitrary bijection $X \cong X$. Morphisms obviously preserves the bijection.

In this topos, the terminal object is the one-element set with the unique bijection. Consider this object $H$ which is a two-element set equipped with the bijection that swaps the two elements. There are no morphisms $1 \to H$ because it would produce a fixpoint of the bijection when there are none. However $H \to 1$ is epi, because the underlying map between sets are epi.

However, your thoughts about the $\mathsf{Set}^2$ thing probably doesn't work because it doesn't form a topos. Also it's evil to consider these unions or intersections of sets without a fixed universal set.

Answer 2

For a geometric example, consider the topos $\operatorname{Sh}(S^1)$ and the "Mobius band" vector bundle on $S^1$. Then if $X$ is the sheaf of everywhere-nonzero continuous sections of that vector bundle, every stalk of $X$ is nonempty, which implies that $X \to 1$ is an epimorphism. On the other hand, $X$ has no global sections $1 \to X$.

More generally, if $\mathscr{F}$ is a sheaf of abelian groups on a topological space $X$, then we have a bijection between $H^1(X, \mathscr{F})$ and the isomorphism classes of $\mathscr{F}$-torsors -- where an object of the latter is a sheaf of sets with an action of $\mathscr{F}$, which locally is isomorphic to $\mathscr{F}$ with left multiplication. (There might be some niceness conditions required for this; I'm not certain, though. In any case, it should hold for Cech cohomology $H^1(X, \mathscr{F})$.) Thus, if $\mathscr{F}$ has nonempty stalks at each point, but $H^1(X, \mathscr{F}) \ne 0$, then there are corresponding examples of $\mathscr{F}$-torsors with no global sections (it is easy to show that an $\mathscr{F}$-torsor has a global section if and only if it is globally isomorphic to $\mathscr{F}$).

In the first paragraph, for example, $X$ is a torsor over the multiplicative group of the sheaf of everywhere-nonzero continuous functions on $S^1$.