The dissolution of the locale associated with a frame $F$ is the locale associated with the frame $N(F)$ of nuclei of $F$ (see, e.g., Johnstone, “Stone Spaces” (1982), §2.5). Note that there is a frame monomorphism $F \to N(F)$ taking $a\in F$ to the nucleus $c(a)\colon F\to F, x \mapsto a\vee x$, which defines an epimorphism of locales in the other direction.
Now if $\mathbf{T}$ is a topos, since $\mathbf{T}$ is equivalent to the category of internal sheaves on the subobject classifier $\Omega$ of $\mathbf{T}$ (or, more precisely, on the internal locale associated to $\Omega$, which is merely the discrete topology on a singleton), I am inclined to define the “dissolution” of $\mathbf{T}$ as the category of sheaves on $N(\Omega)$ or, more precisely, on the dissolution of the discrete topology on a singleton. Note that $N(\Omega)$ can be seen as the “object of Lawvere-Tierney topologies” on $\mathbf{T}$.
Does this notion already exist in the literature (perhaps under a different name)?
Are there perhaps alternative definitions (either alternative ways of rephrasing the definition I gave, or genuinely different notions which may compete with it for the name) of “dissolution” of a topos?