# What's the localic reflection of a presheaf topos?

$$\newcommand{\Psh}{\operatorname{Psh}} \newcommand{\Sh}{\operatorname{Sh}} \newcommand{\O}{{\mathcal{O}}}$$ 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 $$\O(X) = \O(Y) / j$$.

1. Who's $$j$$?

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 left exact left 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, $$\downarrow\! \bigvee V$$.