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.
Sublocales can also be defined for formal topologies.
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.)
(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 (2.1) 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.
Steven Vickers, Sublocales in formal topology. The Journal of Symbolic Logic, Volume 72, Issue 2, June 2007, pp. 463 - 482 (doi:10.2178/jsl/1185803619)
Last revised on May 23, 2023 at 22:58:02. See the history of this page for a list of all contributions to it.