Internal presheaves on the subobject classifier

Question

Let $E$ be a topos and let $\Omega_E$ be its subobject classifier. We can consider $\Omega$ as an internal poset in $E$, and thus we may form the $E$-topos of internal presheaves $\hat{\Omega}_E := [\Omega_E^{\text{op}},E]$. This feels like a "fatter" version of the relative Sierpinski $E$-topos $\mathbb{S}_E := [\cdot\!\to\!\cdot,E]$, but I don't know what it is geometrically. (Indeed, when $E$ is boolean, I suppose this is exactly the Sierpinski $E$-topos.)

I encountered this topos through essentially domain-theoretic considerations; in particular, when developing domain theory within a topos $E$, the partial map classifier monad of the topos lifts to a lifting monad $L$ on internal dcpos. When you consider $L(1) = \Omega_E$, you have what should be considered a Sierpinski space in the sense of internal domain theory and denotational semantics. But taking $E$-valued sheaves on this "Sierpinski $E$-dcpo", I think we would not get the actual Sierpinski $E$-topos, but rather (I suppose), the rather strange topos that I opened the question with.

I am wondering whether anyone has some experience or insight on this topos, and am curious about the following questions:

1. Is there a description of the $E$-geometric theory that $\hat{\Omega}_E$ classifies?
2. There is an essential geometric morphism of $E$-topoi $\mathbb{S}\to \hat{\Omega}_E$ corresponding to the inclusion $2\hookrightarrow\Omega_E$. What other properties does this morphism have?
3. Does this topos occur anywhere in the literature?

Answer 1

I'm not sure this constitute an answer to your question, but from what you are telling of your motivation it very well might be, and in any case it was too long to be a comment.

First - I don't think there is a good answer to your questions 1 and 2. The problem is that considering $\Omega$ as just a poset is a very "non-geometric" thing to do (in the sense of geometric logic) in fact this is somehow the typical non-geometric construction. So asking what its geometric properties are will probably not lead anywhere.

As such, I don't think there is any better description of this topos than the tautological one, that it classifies the ideals of $\Omega_E$, in the sense that a morphism $T \to \widehat{\Omega_E}$ classifies pairs of a morphism $f:T \to E$ and of an ideal of $f^*(\Omega_E)$. The non-geometric nature of $\Omega$ (when considered as a poset) means that $f^*(\Omega_E)$ can be pretty much anything, it has a comparison map to $\Omega_T$ of course, but that just means it has a distinguished subobject - so that not really an interesting observation.

However - there is another construction you can make that seems to fit what you are talking about more closely and produce a much nicer result. The thing is $\Omega$ might not be geometric when seen as a "set" or "poset", but it is geometric when seen with more structure - for example, it is obviously geometric when considered as a frame - but what might be more interesting to you given what you are saying - it is geometric when considered as a DCPO. (what I mean here is that if $f^\sharp$ denotes the left adjoint to $f_*$ acting on the categories of DCPO, then $f^\sharp \Omega_E = \Omega_T$). This is because $\Omega$ is the ind-completion of $\{0 < 1\}$.

So, if you really treat $\Omega$ as a DCPO you'll get much better results. This means you can't look at general presheaves on $\Omega$ though. But there is another way to get a topos out of a DCPO that will be geometric: You can use what Ivan Di liberty and myself have called the "Scott topos" (see here, and also Ivan Di Liberti's thesis and paper).

In Short, given a DCPO $P$, defines $S(P)$ as the category of (covariant) functor $P \to Set$ that preserves directed colimits. Then $S(P)$ is a (localic) topos, and you have a DCPO morphism from $P$ to the poset (DCPO) of points of $S(P)$. The Scott topos is a geometric construction on DCPO in the sense that the Scott topos $S(f^\sharp P)$ identifies with the pullback of the topos $S(P)$.

Moreover, in your present situation, as $\Omega$ is the ind-completion of {0<1} the Scott topos $S(\Omega)$ is exactly the Sierpinski topos.