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.
There are multiple equivalent ways to formally define this concept.
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.
Equivalently, a sublocale of $L$ can be described as a morphism of frames $L\to S$ whose underlying map of sets is surjective.
Given a regular monomorphism $i\colon S\to L$, it induces an adjunction $f^*\dashv f_*$ between the underlying posets. The direct image map
is an injection on underlying sets such that if we equip its image $f_*(S)$ with the induced order, it becomes a locale itself, and the inclusion $f_*(S)\to L$ is a localic map, i.e., it has a left adjoint that preserves finite meets.
Thus, sublocales of $L$ can be defined as subsets $S$ of $L$ such that $S$ is a locale once we equip it with the order induced from $L$, and the left adjoint to the inclusion $S\to L$ exists and preserves finite meets. (Proposition 2.2 in Picado and Pultr PP.)
(The inclusion $S\to L$ need not preserve joins, so in particular, the lattice structure on $S$ may be different from that of $L$, only the ordering is the same.)
Equivalently, sublocales of $L$ can also be described (\cite[2.1]{PP}) as subsets $S$ of $L$ that are closed under all meets and for any $s\in S$ and $x\in L$, we have $(x\to s)\in S$.
Given a surjective morphism of frames $h\colon L\to S$, the subset $h_*(S)$ is a sublocale of $L$.
Given a subset $S$ of a locale $L$ that is a sublocale, the left adjoint $j^*\colon L\to S$ to the inclusion $j_*\colon S\to L$ is a surjective morphism 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:
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 $j$.
Given a nucleus $j$ on a frame $L$, we construct a congruence on $L$ given by the equivalence relation $E$ consisting of all pairs $(x,y)$ ($x,y\in L$) such that $j(x)=j(y)$. The quotient map $L\to L/E$ is a surjective morphism of frames.
Vice versa, given a congruence $E$ on $L$, define $j(x)=\bigvee_{y\colon (x,y)\in E}y$. Then $j$ is the nucleus corresponding to the sublocale $L\to L/E$.
Given a nucleus $j$, we can restrict its codomain to its set-theoretical image, obtaining a surjective morphism of frames $L\to j(L)$.
Vice versa, given a surjective morphism of frames $h^*\colon L\to S$, the composition $h_* h^*\colon L\to L$ is a nucleus on $L$.
Given a nucleus $j\colon L\to L$, the subset $j(L)$ is a sublocale.
Vice versa, given a subset $S\subset L$ that is a sublocale, the corresponding nucleus $j\colon L\to L$ is given by the formula $j(a)=\bigwedge\{s\in S\mid a\le s\}$.
Of course, every locale $L$ is a sublocale of itself. The corresponding nucleus is given by
so every open is preserved in this sublocale. Conversely, every locale has an empty sublocale, given by
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 $U'$ (or any other notation for a complement), given by
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
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 points of $L$.)
The underlying locale of a topological space will typically have many sublocales that are not spatial, i.e., do not come from any topological space.
However, any subset $S$ of a topological space $X$ can be equipped with the induced topology, which turns it into a subspace, whose underlying locale is a sublocale of $\Omega(X)$. This can be most easily seen in the language of surjective morphisms of frames, since open sets in the induced topology are by definition intersections of $S$ and an open subset of $X$.
Thus, we have a canonical map from the set of (topological) subspaces of $X$ to the set of spatial sublocales of $\Omega X$.
This map is a bijection if $X$ satisfies the separation axiom $T_D$, e.g., is a Hausdorff space (VI.1.2 in PP).
This map is a surjection if $X$ is a sober space (Proposition VI.2.2.1 in PP).
Stone Spaces, II.2
Jorge Picado, Aleš Pultr, Frames and Locales.
Last revised on March 27, 2020 at 07:48:31. See the history of this page for a list of all contributions to it.