Connections between $0$-toposes and $1$-toposes (Grothendieck and elementary) This is a crosspost from math.stackexchange.
A Grothendieck $0$-topos is the same as a frame (see here). Another relationship between Grothendieck toposes and frames is the following: whenever $\mathcal O$ is a frame, then $\mathrm{Sh}(\mathcal O)$, the category of sheaves on $\mathcal O$, is a Grothendieck topos.
A elementary $0$-topos is the same as a Heyting algebra (see here). Another relationship between elementary toposes and Heyting algebras is the following: if $\mathcal E$ is an elementary topos and $X\in \mathcal E$ is an object, then $\mathrm{Sub}_\mathcal E(X)$ is a Heyting algebra.
These facts suggest to me these question:

*

*If $H$ is a Heyting algebra, does $H$ induce an elementary topos (in the same way a frame $\mathcal O$ induces the Grothendieck topos $\mathrm{Sh}(\mathcal O)$)?


*If $\mathcal E$ is a Grothendieck topos and $X\in\mathcal E$ an object, is $\mathrm{Sub}_\mathcal E(X)$ a frame?


*Do these connections hold in general for (Grothendieck or elementary) $n$-toposes? That is, whenever $\mathcal E$ is a (Grothendieck or elementary) $n$-topos and $X\in\mathcal E$, then $\mathrm{Sub}_\mathcal E(X)$ is a (Grothendieck or elementary) $n-1$-topos, and whenever $\mathcal E$ is a (Grothendieck or elementary) $n$-topos, then there's an induced $n+1$-topos $\mathrm{Sh}(\mathcal E)$?
The second question was already answered by Mark Kamsma: yes.
I want to add the following question: what's the intuitive reason that a 0-topos being a frame / Heyting algebra implies that each frame induces a topos and each collection of subobjects is a Heyting algebra? Is this just a coincidence?
 A: *

*Yes and no.  As Jonas Frey pointed out on MSE, you can take the category of finite sheaves on any Heyting algebra $H$ to get an elementary topos.  However, unlike in the case of frames and Grothendieck toposes, the Heyting algebra $H$ will not in general coincide with the Heyting algebra of subterminal objects in this elementary topos.  Roughly speaking, the topos of finite sheaves doesn't "know" anything about the Heyting structure of $H$, only its underlying distributive lattice (whereas for a frame, the Heyting structure is recoverable from the infinite distributive law and the adjoint functor theorem).  It might still be an open question whether any Heyting algebra can occur as the Heyting algebra of subterminal objects in an elementary topos; at least, I believe it was open for a while.


*Yes, as you say.


*In the Grothendieck case, yes, sort of.  On one hand, if $E$ is a Grothendieck $n$-topos and $X\in E$, then the category of $(n-2)$-truncated objects in $E/X$ is an $(n-1)$-topos; but it only really makes sense to call these "subobjects" when $n=1$.  On the other hand, if $E$ is a Grothendieck $n$-topos, you can identify it with the category of $n$-sheaves on some site $C$, and then the category of $(n+1)$-sheaves on $C$ will be an $(n+1)$-topos whose underlying category of $(n-1)$-truncated objects in $E$.
