A topos where every object is internally projective but not every object is projective

Question

An object $P$ in a topos $\mathcal{E}$ is said to be projective if $Hom_{\mathcal{E}}(p,-)$ preserves epis, internally projective if $(-)^P$ preserves epis.

Can anyone give an example of a topos where every object is internally projective but not every object is projective? In general, what is the condition for a presheaf topos satisfying AC or IAC?

Answer 1

The question was asked on MathOverflow, with the example given in the answer: https://mathoverflow.net/q/299591 And the condition for a presheaf topos satisfying IAC or AC was given in the comment. I summarised the comment below and gave a proof for it.

Assuming the category $\mathbf{Sets}$ satisfies AC, for a presheaf topos $\mathcal{ E}=\mathbf{Sets}^{\mathbf{C}^{op}}$, $\mathcal{E}$ satisfies IAC iff $\mathbf{C}$ is a groupoid, and $\mathcal{E}$ satisfies AC iff $\mathbf{C}$ is discrete.

Proof. If $\mathcal{ E}$ satisfies IAC, then by Diaconescu's result which states that if an elementary topos $\mathcal{ E}$ satisfies IAC then it is Boolean. Also, being Boolean is equivalent to that $\mathbf{C}$ is a groupoid. Conversely, assume $\mathbf{C}$ is a groupoid, then every arrow $f: C\to C'$ in $\mathbf{C}$ is an isomorphism. Thus the following diagram $\require{AMScd}$

\begin{CD} {Hom(C,C)\times PC} @>{\alpha_{C}}>> QC \\ @VHom(f,C)\times PfVV @VVQfV\\ {Hom(C',C)\times PC'} @>{\alpha_{C'}}>> QC' \end{CD} gives the isomorphism $Q^P(C)\cong Hom_{\mathcal{ E}}(PC,QC)$ which is the set of all functions from $PC$ to $QC$. Also, in the functor category, an arrow being epic (monic) is equivalent to being pointwise epic (monic). Thus it can be seen that $\mathcal{ E}$ satisfies IAC.

If $\mathcal{ E}$ satisfies AC, consider the section of arrow $P\to 1$ and notice that the terminal object $1\in\mathcal{E}$ sends every object $C\in\mathbf{C}$ to $\{0\}$ and every arrow $f$ in $\mathbf{C}$ to the identity $id_0$. This gives that the $\mathbf{C} $ has to be discrete. Conversely, if $\mathbf{C}$ is discrete, then $\mathbf{C}$ is equivalent to a set $I$ and every object $X\in\mathcal{ E}$ is a family of indexed sets $X=\{X_i\mid i\in I\}$. Any epi $f:X\twoheadrightarrow Y$ is a family of indexed surjections $f=\{f_i:X_i\twoheadrightarrow Y_i\mid i\in I\}$ which has a section. So $\mathcal{ E}$ satisfies AC.

Answer 2

In a topos where every object is internally projective, every object is projective iff the terminal object $1$ is projective iff the global sections functor $\text{Hom}(1, -)$ preserves epis.

Now, you can verify that in the topos of $G$-sets, $G$ a group (which is a presheaf topos), every object is internally projective (because the internal hom is just $\text{Hom}_{\text{Set}}(X, Y)$ equipped with the usual conjugation $G$-action, and sets are projective), but the fixed point functor $\text{Hom}(1, -)$ preserves epis iff $G$ is trivial; for example the unique map $G \to 1$ is an epimorphism but $\text{Hom}(1, G) = \emptyset$ unless $G$ is trivial.