Preimage of a sublocale by a morphism of locales: description by nucleus?

Question

For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs directly to the question).

A frame is a lattice admitting arbitrary joins ($\bigvee$) and such that finite meets ($\wedge$) distribute over these. (In particular, it has a Heyting operation defined by $(U\Rrightarrow W) := \bigvee_{U\land V\leq W} V$.) Frame homomorphisms are maps preserving finite meets (including the top element) and arbitrary joins (but not necessarily the Heyting operation). The category of locales is the opposite of the category of frames: we write $\mathcal{O}(X)$ for the frame associated to a locale $X$ and we call it its “frame of opens”. A morphism $f\colon X\to Y$ of locales is thus defined by a morphism $f^*\colon \mathcal{O}(Y) \to \mathcal{O}(X)$ of frames, but it can just as well be defined by the order-theoretic right adjoint $f_*\colon \mathcal{O}(X) \to \mathcal{O}(Y)$ (that is, $f_*(V) = \bigvee_{f^*(U)\leq V} U$: this is generally not a morphism of frames, but as a right adjoint, it preserves arbitrary meets; these $f_*$ are sometimes called “localic maps”).

An inclusion of locales is a morphism $i\colon Z\to X$ such that $i^*\colon \mathcal{O}(X) \to \mathcal{O}(Z)$ is surjective (i.e., $\mathcal{O}(Z)$ is a quotient frame of $\mathcal{O}(X)$). A nucleus on $\mathcal{O}(X)$ is a map $j\colon \mathcal{O}(X) \to \mathcal{O}(X)$ which satisfies ① $j(U\wedge V) = j(U)\wedge j(V)$ (from which it follows that $j$ is order-preserving), ② $U \leq j(U)$, and ③ $j(j(U)) = j(U)$. A sublocale (or perhaps better worded, “sublocalic subset”) of $\mathcal{O}(X)$ is a subset $S \subseteq \mathcal{O}(X)$ which is closed under arbitrary meets and which is an exponential ideal in the sense that $(U\Rrightarrow W) \in S$ whenever $W\in S$ and $U\in\mathcal{O}(X)$. These are all equivalent data: there are canonical bijections between inclusions-up-to-isomorphism (of $Z$ over $X$), quotient frames of $\mathcal{O}(X)$, nuclei on $\mathcal{O}(X)$ and sublocales of $\mathcal{O}(X)$. (For example, to $i\colon Z\to X$ we associate the nucleus $j := i_* i^*$, and to a nucleus $j$ on $\mathcal{O}(X)$ we associate the sublocale $S := j(\mathcal{O}(X))$ which is the set-theoretic image of $j$ and which is also its fixset $\{U \in \mathcal{O}(X) : j(U)=U\}$.)

If $f\colon X \to Y$ is a morphism of locales, and $S \subseteq \mathcal{O}(Y)$ a sublocale of $Y$, we define the preimage of $S$ by $f$ [see: Picado & Pultr, Frames and Locales, III.4.2] to be the smallest sublocale of $\mathcal{O}(X)$ containing $(f_*)^{-1}(S) := \{V \in \mathcal{O}(X) : f_*(V) \in S\}$ (this set is already closed under arbitrary meets, so the issue is that of the Heyting operation).

This description of the preimage operation takes place on the “sublocale” description. But I would like to see how it works at the “nucleus” level. Hence:

QUESTION: Given $f\colon X \to Y$ is a morphism of locales, and $j$ a nucleus on $\mathcal{O}(Y)$, is there a way to directly describe the nucleus corresponding to the preimage (of the sublocale corresponding to $j$) by $f$? Is it, for example, given by $U \mapsto U \vee f^* \circ j\circ f_*(U)$ (which is at least superficially plausible) or something of the sort?

Answer 1

Here is I think a counter-example to the precise proposed formula in the question.

Take $X= \mathbb{Q}$ with the discrete topology, with $Y = \mathbb{R}$ withe its usual topoly and the map $f:X \to Y$ to be the inclusion maps. and take $j$ the nucleus for the inclusion $U \subset \mathbb{R}$ for $U$ a dense open subset not containing $\mathbb{Q}$. For example $U = \mathbb{R}-\mathbb{Z}$, that is

$$j(X)= \left( U \Rightarrow X \right) = (X \cup \mathbb{Z})^\circ.$$

The pullback is $(\mathbb{Q} - \mathbb{Z}) \subset \mathbb{Q}$, whose nuclei is $j'(A) = A \cup \mathbb{Z}$.

Take $S = \varnothing$. We get $j'(\varnothing ) = \mathbb{Z}$

The formula proposed in the question would give:

$$ j''(\varnothing) = \varnothing \cup f^* ((\mathbb{Z} \cup f_* (\varnothing))^\circ) = f^* (\mathbb{Z}^\circ) = f^* (\varnothing) = \varnothing \neq j'(\varnothing)$$

(where $f_*(\varnothing) = \varnothing $ because $\mathbb{Q}$ is dense)

I don't think it is possible to give a formula using only finitary or even "bounded" operations on the frames. But, if we relax this, the following work:

Call "pre-nuclei" a function $j_0:O(Y) \to O(Y)$ which only satisfies $x \leqslant j_0(x)$ and $j_0(a \cap b) = j_0(a ) \cap j_0(b)$.

Then there is a formula for the smallest nuclei above $j_0$:

you just need to define $j_{\alpha}$ for an ordinal $\alpha$ by $j_{\alpha+1} = j_0 \circ j_\alpha$ and

$$j_\lambda(x) = \bigvee_{\beta < \lambda} j_\beta(x)$$

At some point $j_\lambda$ stabilize (at worst at $\lambda$ the cardinality of the frame - but probably much before). And it is easy to see that each $j_\beta$ is a pre-nuclei, $j_0 j_\lambda = j_\lambda$ and then by induction that $j_\lambda j_\lambda = j_\lambda$ so that $j_\lambda$ is a nuclei, and the smallest nuclei above $j_0$.

More generally, a directed union of pre-nuclei is a pre-nuclei, and an intersection of pre-nuclei is a pre-nuclei. So you can also use this to compute supremums of familly of nuclei for example.

In the case of the pullback of frame, you can start with

$$ j_0 (S) = \bigvee_{a \in O(X)} \left( f^*(a) \Rightarrow S \vee f^*(ja) \right) $$

This work because for $A \subset B$ in $O(X)$, $j(S) = A \Rightarrow (S \vee B)$ is the nuclei corresponding to the localization at $A \subset B$, so the above $j_0$ is the pre-nuclei corresponding to the localization at all the $f^* U \subset f^* j U$. (Note that the union is directed).

Of course for nice sublocale, like open or closed or even the once that are obtained as localization at a finite number of inclusion $A_i \subset B_i$ you can give simpler finitary formula, but in the most general case I don't think you can even give a uniform bound on the length of the required iteration.