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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?

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  • $\begingroup$ This is interesting; I for one haven’t come across it before. An obvious “sanity check” question is whether it really generalises the existing localic definition: when $\newcommand{\T}{\mathbf{T}}\newcommand{\Sh}{\mathrm{Sh}}\T=\Sh(X)$, do we have $\Sh_{\T}(N(\T)) \simeq \Sh(N(X))$? This seems not quite obvious to me — I guess it comes down to comparing the totalisation of the internal frame $\int_{\mathcal{O}(X)}N(\Omega_\T)$ with the externally defined $N(\mathcal{O}(X))$. $\endgroup$ Feb 28, 2023 at 9:02
  • $\begingroup$ @PeterLeFanuLumsdaine Indeed, I wrote the question assuming this point was obvious, and later realized it wasn't. I need to think about it! $\endgroup$
    – Gro-Tsen
    Feb 28, 2023 at 11:06

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I'm not aware of litterature on this, but this is something I have thought about several years ago and never ended-up using or publishing. What is below is me trying to remind myself how it works - unfortunately without my notes.

So,I think a good way to understand the dissolution of a topos is in terms of its universal property. First let me recall how it works from locales (This is done in C1 in the elephant, and I think also in Stone in space but I can't check now)

The frame homomorphism $X \to N(X)$ is universal for frame homomorphism $X \to Y$ that sends all elements of $X$ to complemented element of $Y$.

Note that it is also easy to construct "partial dissolution" where you start form a set $S \subset F$ and construct $N_S(F)$ which is universal for making the element of $S$ complemented. THis can be for example constructed as a subframe of N(F).

The analogous construction for a topos $T$, is:

Definition For T a topos, and I a collection of subobject inclusion $(U_i \subset V_i)_{i\in I}$ in $T$, the dissolution of $T$ at $I$ is (if it exists) a topos $N_I(T)$ endowed with an algebraic morphism $T \to N_I(T)$ which is universal for algebraic morphisms $T \to E$ sending all $U_i \subset V_i$ to complemented subobject inclusion in $E$.

By "algebraic morphism" I just mean "Geometric morphism" but in the $f^*$ direction.

It is easy to construct such $N_I(T)$ when $T$ is a Grothendieck topos and $I$ is a set as a classyfing topos: represent $T$ as a classyfing topos in a way so that all the $V_i$ are sorts and all the $U_i$ are definable subsets and then add sort $C_i$ for the complement of $U_i$ with axioms that forces $C_i$ to be the complement of $U_i$ in $V_i$.

An easy but key lemma is the following:

Lemma Let $T$ be a Grothendieck topos, $(X_i \to Y)_{i\in I}$ be a covering family, $S \subset Y$ a subobject and $S_i = X_i \times_Y S$. Let $f:T \to E$ be an algebraic morphism. each each $f(S_i) \subset f(X_i)$ is complemented then $f(S) \subset f(Y)$ is complemented.

It follows that

Proposition: Let $T$ be a Grothendieck topos, then the dissolution $N(T)$ at all subobject inclusion in $T$ exists.

Proof Take a site of a definition for $T$ and construct the dissolution of $T$ at all subobject inclusion $S \subset c$ where $c$ is representable.

Proposition: $N(Sh(X)) = Sh(N(X))$

Proof Sh(N(X)) is clearly universal for making all $V \subset 1$ complemented in $Sh(X)$. This implies that it also makes all $U \subset V$ for $U,V$ subterminal objects complemented. But as in $Sh(X)$ every object is covered by subterminal objects, this imply that it makes all subobject inclusion complemented because of the lemma.

Finally:

Proposition The dissolution $N(T)$ of a Grothendieck topos can also be constructed internally as sheaves over the internal frame $N(\Omega)$ as suggested in the question.

Proof Reasoning internally in $T$, an algebraic morphism $Sh_T(N(\Omega)) \to E$ for a $T$-topos $E$ is the same as an (so... the unique) algebaic morphism $* \to E$ such that internally it sends every sub-object inclusion to a complemented one. Expressing this externally, $E$ corresponds to a topos with an algebraic morphism $T \to E$ and an extension to $Sh_T(N(\Omega))$ eaxclty means that it sends every subobject to a complemented one, which is exactly the universal property of $N(T)$.

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