sublocale

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 $Sub Top(X) \hookrightarrow Sub Loc(X)$ which, while injective, is usually far from surjective.

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:

- $j(U \cap V) = j(U) \cap j(V)$,
- $U \subseteq j(U)$,
- $j(j(U)) = j(U)$.

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 $j$.

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

$j_L(U) \coloneqq U ,$

so every open is preserved in this sublocale. Conversely, every locale has an empty sublocale, given by

$j_\empty(U) \coloneqq L .$

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

$j_U(V) \coloneqq U \Rightarrow V ,$

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

$j_{U'}(V) \coloneqq U \cup V ,$

so $j_{U'}(V)$ is the largest open which agrees with $V$ except on $U$. If $L$ is topological, then every open sublocale of $L$ is also topological; the same goes for closed sublocales, assuming excluded middle. Even in constructive mathematics, however, open and closed sublocales are complements in the lattice of sublocales.

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

$j_{\neg\neg}(U) \coloneqq \neg{\neg{U}} .$

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; in fact, its only points are the isolated point?s of $L$.)

*Stone Spaces*, II.2

Revised on December 4, 2016 04:03:58
by Toby Bartels
(108.167.41.14)