Let $C$ be a topos with subobject classifier $\Omega$. Let $F$ be the endofunctor $x \mapsto \Omega^{\Omega^x}$ on $C$. Does there exist $C$ such that $F$ has an initial algebra? What if $\Omega$ is replaced with the coproduct $2 = 1 \sqcup 1$ of two copies of the terminal object? (What if it is replaced with any object that is not the terminal object?) 


The motivation for this question is a somewhat long story, but if someone wants I can edit it in.


Note that if $C = \text{Set}$ then this is impossible by Cantor's theorem (and Lambek's theorem). I'm worried that the [Lawvere fixed point theorem](https://mathoverflow.net/questions/127045/fixed-point-theorems/127103#127103) can shoot down this question as well. 



Does this kind of endofunctor ever have an initial algebra? 
By Lambek's theorem, any initial algebra for an endofunctor $F$ has the property that its structural morphism $\alpha : F A \to A$ is an isomorphism. So we seek an object $A$ such that $P P A = \Omega^{\Omega^A} \cong A$.


However:


**Proposition.** If a topos $\mathcal{E}$ contains an object $A$ such that $P A$ is a subquotient of $A$, then $\mathcal{E}$ is a degenerate topos (i.e. $0 \cong 1$).


*Proof.* This is Proposition D4.1.8 in [Johnstone, *Sketches of an elephant*]. 　◻


So, suppose that $A \cong P P A$. Clearly, $A$ is a subobject of $P A$ via the singleton operator $\lbrace \cdot \rbrace\_A : A \to P A$, so this isomorphism exhibits $P P A$ as a subquotient of $P A$, which by the proposition implies our topos is degenerate.


That said, $\Omega$ is a bit special. The general form of Cantor's theorem for function-sets is known to fail in toposes; for example, in the effective topos, if $N$ is the natural numbers object, then $N^N$ is a subquotient of $N$. There are also cartesian closed categories (hence toposes via Yoneda) containing an object $A$ such that $A \cong A^A$ (and hence $A \cong A^{A^A}$).


