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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, $\downarrow\! \bigvee V$.

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, $\downarrow\! \bigvee V$.

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

Opens of $Y$ are sieves on $X$, or rather I should say on $\mathcal{O}(X)$, i.e. the collection of open subsets $V \subset \mathcal{O}(X)$ such that $v \in V$ and $u \leqslant v \Rightarrow u \in V$.

More generally if $P$ is a poset, a subterminal presheaf on $P$ is a functor $P \to \{ \emptyset,1\}$ 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 open corresponds to the principal sieve, i.e. the one of the form $\downarrow v = \{ u \in \mathcal{O}(X) | u \leqslant v \}$.

The nucleus $j$ is hence the reflection on this principal sieve: it takes a general sieve $V \subset \mathcal{O}(X)$ and sends it to $\downarrow v$ where $v = \bigcup V$ is the supremum/the union of the elements of $V$. It is the smallest principal ideal containing $V$.

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, $\downarrow\! \bigvee V$.

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

Opens of $Y$ are sieves on $X$, or rather I should say on $\mathcal{O}(X)$, i.e. colection of open subsets $V \subset \mathcal{O}(X)$ such that $v \in V$ and $u \leqslant v \Rightarrow u \in V$.

More generally if $P$ is a poset, a subterminal presheaves on $P$, is a functor $P \to \{ \emptyset,1\}$ 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 identifies $X$ as a sublocale of $Y$, its open corresponds to the principale sieves, i.e. the one of the form $\downarrow v = \{ u \in \mathcal{O}(X) | u \leqslant v \}$.

The nucleous $j$ is hence the reflection on these principale sieves: it takes a general sieve $V \subset \mathcal{O}(X)$ and send it to $\downarrow v$ where $v = \bigcup V$ is the supremum/the union of the elements of $V$. It is the smallest principal ideal containing $V$.

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

Opens of $Y$ are sieves on $X$, or rather I should say on $\mathcal{O}(X)$, i.e. the collection of open subsets $V \subset \mathcal{O}(X)$ such that $v \in V$ and $u \leqslant v \Rightarrow u \in V$.

More generally if $P$ is a poset, a subterminal presheaf on $P$ is a functor $P \to \{ \emptyset,1\}$ 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 open corresponds to the principal sieve, i.e. the one of the form $\downarrow v = \{ u \in \mathcal{O}(X) | u \leqslant v \}$.

The nucleus $j$ is hence the reflection on this principal sieve: it takes a general sieve $V \subset \mathcal{O}(X)$ and sends it to $\downarrow v$ where $v = \bigcup V$ is the supremum/the union of the elements of $V$. It is the smallest principal ideal containing $V$.