What's the localic reflection of a presheaf topos?

Question

$\newcommand{\Psh}{\operatorname{Psh}} \newcommand{\Sh}{\operatorname{Sh}} \newcommand{\O}{{\mathcal{O}}}$ Let $X$ be a locale, $\O(X)$ the corresponding frame.

1. What's the localic reflection of $\Psh X$?

We know that $$ \O(X) \cong \mathrm{Sub}_{\Sh X}(1) $$ Call $Y = \mathrm{Sub}_{\Psh X}(1)$ the localic reflection of $\Psh X$. One has $\Psh X \simeq \Sh Y$.

Since $\Sh X$ is a subtopos of $\Psh X$, $X$ should be a sublocale of $Y$, i.e. there should be a nucleus $j$ on $\O(Y)$ such that $\O(X) = \O(Y) / j$.

2. Who's $j$?

Answer 1

I'm writing $\mathcal{O}(X)$ for the frame corresponding to $X$.

Opens of $Y$ are sieves on $X$, i.e. the collection of open subsets $V \subset \mathcal{O}(X)$ such that $v \in V$ and $u \leqslant v \Rightarrow u \in V$. Thus $Y$ is the locale of downward closed subsets of $\mathcal O(X)$.

More generally if $P$ is a poset, a subterminal presheaf on $P$ is a functor $P \to \{ 0,1\}$ and the set of $p \in P$ sent to $1$ is a sieve on $P$. This induces a bijection between sieves and subterminal presheaves.

When you identify $X$ as a sublocale of $Y$, its opens correspond to principal sieves, i.e. the ones of the form $\downarrow\! v = \{ u \in \mathcal{O}(X) | u \leqslant v \}$.

Thus the inclusion of locales $X \hookrightarrow Y$ is given by the quotient of frames $\mathcal{O}(Y) \to \mathcal{O}(X)$ defined as $V \mapsto \bigvee V$ and whose right adjoint is $v \mapsto {\downarrow\! v}$.

The nucleus $j$ hence takes a general sieve $V \subset \mathcal{O}(X)$ and sends it to the smallest principal ideal containing $V$, that is, $\downarrow\! \bigvee V$.