# 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 $\mathrm{Sub}\mathrm{Top}\left(X\right)↪\mathrm{Sub}\mathrm{Loc}\left(X\right)$ 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. $j\left(U\cap V\right)=j\left(U\right)\cap j\left(V\right)$,
2. $U\subseteq j\left(U\right)$,
3. $j\left(j\left(U\right)\right)=j\left(U\right)$.

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

${j}_{L}\left(U\right)≔U,$j_L(U) \coloneqq U ,

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

${j}_{U}\left(V\right)≔U⇒V,$j_U(V) \coloneqq U \Rightarrow V ,

so ${j}_{U}\left(V\right)$ 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

${j}_{U\prime }\left(V\right)≔U\cup V,$j_{U'}(V) \coloneqq U \cup V ,

so ${j}_{U\prime }\left(V\right)$ 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 ${L}_{¬¬}$, is given by

${j}_{¬¬}\left(U\right)≔¬¬U.$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, ${L}_{¬¬}$ is rarely topological.)

## References

Revised on July 2, 2010 21:18:21 by Toby Bartels (75.88.100.190)