Sublocales

Idea

A sublocale is a subspace of a locale.

It is important to understand that, even for a topological locale $X$ (which can be identified with a sober topological space), most sublocales of $X$ are not topological. Specifically, we have an inclusion function $\mathrm{Sub}\mathrm{Top}(X)\hookrightarrow \mathrm{Sub}\mathrm{Loc}(X)$ which, while injective, is usually far from surjective.

Definitions

Let $L$ be a locale, which (as an object) is the same as a frame.

A sublocale of $L$ is a regular subobject of $L$ in Loc, the category of locales. Equivalently, it is a regular quotient of $L$ in Frm, the category of frames.

A sublocale is given precisely by a nucleus on the underlying frame. This is a function $j$ from the opens of $L$ to the opens of $L$ satisfying the following identities:

1. $j(U\cap V)=j(U)\cap j(V)$,
2. $U\subseteq j(U)$,
3. $j(j(U))=j(U)$.

In other words, a sublocale of $L$ is given by a meet-preserving monad on its frame of opens.

The precise reasons why nuclei correspond to quotient frames (and hence to sublocales) is given at nucleus. But the interpretation of the operation $j$ is this: we identify two opens if they ‘agree on the sublocale’. Given an open $U$, there will always be a largest open that is identified with $U$, so we can also describe a subspace of a locale as an operation that maps each open to its largest representative open in the sublocale. This map is the nucleus $j$.

Special cases

Of course, every locale $L$ is a sublocale of itself. The corresponding nucleus is given by

so every open is preserved in this sublocale.

Suppose that $U$ is an open in the locale $L$. Then $U$ defines an open subspace of $L$, also denoted $U$, given by

so $j_U(V)$ is the largest open which agrees with $V$ on $U$. $U$ also defines a closed subspace of $L$, denoted $U\prime$ (or any other notation for a complement), given by

so $j_{U\prime }(V)$ is the largest open which agrees with $V$ except on $U$. If $L$ is topological, then every open or closed sublocale of $L$ is also topological.

The double negation sublocale of $L$, denoted $L_{\neg \neg }$, is given by

This is always a dense subspace; in fact, it is the smallest dense sublocale of $L$. (As such, even when $L$ is topological, $L_{\neg \neg }$ is rarely topological.)

References

• Stone Spaces, II.2