CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A locale is, intuitively, like a topological space that may or may not have enough points (or even any points at all). It contains things we call open subspaces but there may or may not be enough points to distinguish between open subspaces. An open subspace in a locale can be regarded as conveying a bounded amount of information about the (hypothetical) points that it contains.
For example, there is a locale of all surjections from natural numbers (thought of as forming the discrete space $N$) to real numbers (forming the real line $R$). This locale has no points, since there are no such surjections, but it contains many nontrivial open subspaces; these open subspaces are generated by a family parametrised by $n\colon N$ and $x\colon R$; the basic open associated to $n$ and $x$ may be described as $\{f\colon N \twoheadrightarrow R \;|\; f(n) = x\}$.
Every topological space can be regarded as a locale (with some lost information if the space is not sober). The locales arising this way are the topological or spatial locales. Conversely, every locale induces a topology on its set of points, but sometimes a great deal of information is lost; in particular, there are many different locales whose set of points is empty.
One motivation for locales is that since they take the notion of open subspace as basic, with the points (if any) being a derived notion, they are exactly what is needed to define sheaves. The notion of sheaf on a topological space only refers to the open subspaces, rather than the points, so it carries over word-for-word to a definition of sheaves on locales. Moreover, passage from locales to their toposes of sheaves is a full and faithful functor, unlike for topological spaces.
Another advantage of locales is that they are better-behaved than topological spaces in alternative foundations of mathematics, including mathematics without the axiom of choice, more generally constructive mathematics, or mathematics internal to an arbitrary topos. For example, constructively the topological space $[0,1]$ need not be compact, but the locale $[0,1]$ is always compact (in a suitable sense). It follows that the locale $[0,1]$, and hence also the locale $R$ of real numbers, is not necessarily spatial. When it fails to be spatial, because there are “not enough real numbers,” the locale or real numbers is generally a better-behaved object than the topological space of real numbers.
Recall that a frame $A$ is a poset with all joins and all finite meets which satisfies the infinite distributive law:
A frame homomorphism $\phi\colon A\to B$ is a function which preserves finite meets and arbitrary joins. Frames and frame homomorphisms form a category Frm.
Note: By the adjoint functor theorem (AFT) for posets, a frame also has all meets (at least assuming that it is a small set, or more generally that we allow impredicative quantification over it), but a frame homomorphism need not preserve them. Similarly, by the AFT, a frame is automatically a Heyting algebra, but again a frame homomorphism need not preserve the Heyting implication.
By definition, the category Locale of locales is the opposite of the category of frames
That is, a locale $X$ “is” a frame, which we often write as $O(X)$ and call “the frame of open subspaces of $X$”, and a continuous map $f\colon X \to Y$ of locales is a frame homomorphism $f^*\colon O(Y) \to O(X)$. If you think of a frame as an algebraic structure (a lattice satisfying a completeness condition), then this is an example of the duality of space and quantity.
It is also possible to think of a continuous map $f\colon X \to Y$ as a map of posets $f_*\colon O(X) \to O(Y)$: a function that preserves all meets (and therefore is monotone and has a left adjoint $f^*\colon O(Y) \to O(X)$) and such that the left adjoint $f^*$ preserves all finite meets. This has the arrow pointing in the “right” direction, at the cost of a less direct definition. (In predicative mathematics, we must explicitly add into this definition that $f_*$ has a left adjoint; compare the definition of geometric morphism one level up.)
This category is naturally enhanced to a 2-category:
The 2-category Locale has
as morphisms $f\colon X \to Y$ frame homomorphisms $f^*\colon O(Y) \to O(X)$;
a unique 2-morphism $f \Rightarrow g$ whenever for all $U \in O(Y)$ we have $f^*(U) \leq g^*(U)$.
(See for instance Johnstone, C1.4, p. 514.)
Since parallel $2$-morphisms are equal, this 2-category is in fact a (1,2)-category: a Poset-enriched category.
Given a locale $X$, the elements of the frame $O(X)$ are traditionally thought of as being the open subspaces of $X$ and are therefore called the opens (or open subspaces, open parts, or open sublocales) of $X$. However, one may equally well view them as the closed subspaces of $X$ and call them the closeds (or closed subspaces, closed parts, or closed sublocales) of $X$. When viewed as closed subspaces, the opposite containment relation is used; thus $O(X)$ is the frame of opens of $X$, while the opposite poset $O(X)^{op}$ is the coframe of closeds of $X$.
An open subspace may be thought of as a potentially verifiable property of a hypothetical point of the space $X$. Thus we may verify the intersection of two open subspaces by verifying both properties, and we may verify the union of any family of open subspaces by verifying at least one of them; but it may not necessarily make sense to verify the intersection of infinitely many open subspaces, because this would require us to do an infinite amount of work. (The meet of an arbitrary family of open subspaces does exist, but the construction is impredicative, and it does not match the meet within the lattice of all subspaces.)
Dually, a closed subspace may be thought of as a potentially refutable property. Thus we may refute the union of two closed subspaces by refuting both of them, and we may refute the intersection of any family of closed subspaces by refuting at least one of them; but it may not necessarily make sense to refute the union of infinitely many closed subspaces. (Again, the join of an arbitrary family of closed subspaces does not work right.)
If one views an element of $O(X)$ as a subspace of $X$, one usually means to view it as an open subspace, but we have seen that one may also view it as a closed subspace. This is given by two different maps (one covariant and a frame homomorphism, one contravariant and a coframe homomorphism) from $O(X)$ to the lattice of all subspaces of $X$. See sublocale for further discussion.
As a locale, the abstract point is the locale $q$ whose frame of opens is the frame of truth values (classically $\{\bot \lt \top\}$). This is the terminal object in $Locale$, since we must have (for $f\colon X \to 1$) $f^*(\top) = \top_X$ and $f^*(\bot) = \bot_X$ (and most generally $f^*(p) = \bigvee \{\top_X \;|\; p\}$, since $p = \bigvee \{\top \;|\; p\}$).
Given a locale $X$, a concrete point of $X$ may be defined in any of the following equivalent ways:
Definition (3) is simpler than (2), being an element of $O(X)$ satisfying a finitary condition rather than a subset of $O(X)$ satisfying an infinitary condition. However, it doesn't work in constructive mathematics, which provides much (but by no means all) of the motivation for studying locales.
The category Locale$(E)$ of locales internal to a topos $E$ is equivalent to the category of localic geometric morphisms $Sh_{E}(\Sigma) \to E$ in Topos.
See localic geometric morphism for more.
For every locale $X$, the category $Locale(Sh(X))$ of locales internal to the sheaf topos over $X$ is equivalent to the overcategory $Locale/X$
This appears as Johnstone, theorem C1.6.3.
For every morphism of locales $f\colon Y \to X$ the sheaf topos $Sh(Y)$ is equivalent as a topos over $Sh(X)$ to the topos $Sh_{Sh(X)}(\mathcal{I}(Y))$ of internal sheaves over the internal locale $\mathcal{I}(Y) \in Sh(X)$
This appears as Johnstone, scholium C1.6.4.
Every topological space $X$ has a frame of opens $O(X)$, and therefore gives rise to a locale $X_L$. For every continuous function $f\colon X \to Y$ between topological spaces, the inverse image map $f^{-1}\colon O(Y) \to O(X)$ is a frame homomorphism, so $f$ induces a continuous map $f_L\colon X_L \to Y_L$ of locales. Thus we have a functor
$(-)_L\colon$ Top $\to$ Locale.
Conversely, if $X$ is any locale, we define a point of $X$ to be a continuous map $1 \to X$. Here $1$ is the terminal locale, which can be defined as the locale $1_L$ corresponding to the terminal space. Explicitly, we have $O(1) = P(1)$, the powerset of $1$ (the initial frame, the set of truth values, which is $2$ classically or in a Boolean topos). Since a frame homomorphism $O(X) \to P(1)$ is determined by the preimage of $1$ (the true truth value), a point can also be described more explicitly as a completely prime filter: an upwards-closed subset $F$ of $O(X)$ such that $X \in F$ ($X$ denotes the top element of $O(X)$), if $U,V \in F$ then $U \cap V \in F$, and if $\bigcup_i U_i \in F$ then $U_i \in F$ for some $i$.
The elements of $O(X)$ induce a topology on the set of points of $X$ in an obvious way, thereby giving rise to a topological space $X_P$. Any continuous map $f\colon X \to Y$ of locales induces a continuous map $f_P\colon X_P \to Y_P$ of spaces, so we have another functor
$(-)_P\colon Loc \to Top$.
One finds that $(-)_L$ is left adjoint to $(-)_P$.
In fact, this is an idempotent adjunction:
The categories Top of topological spaces and Locale of locales are related by an idempotent adjunction.
This appears for instance as (MacLaneMoerdijk, theorem IX.3 1) or as (Johnston, lemma C.1.2.2).
Therefore the adjunction restricts to an equivalence between the fixed subcategories on either side.
A topological space $X$ with $X \cong X_{L P}$ is called sober.
A locale with $X \cong X_{P L}$ is called spatial or topological; one also says that it has enough points.
The adjunction from prop. 4 exhibits sober topological spaces as a coreflective subcategory of Locale
and this restricts to an equivalence of categories between the full subcategory of locales with enough points, and that of sober topological spaces.
This appears for instance as (MacLaneMoerdijk, corollary IX.3 4).
Consequently, we often identify a sober topological space and the corresponding topological locale.
The notion of Grothendieck topos can be seen as a categorification of the notion of locale, by relating both notions to the notion of lex totality:
A poset $P$ is a frame if and only if the Yoneda embedding
has a left exact left adjoint. (Here the poset $\mathbf{2}$ of truth values is the base of enrichment for posets seen as enriched categories.)
The analogous result for toposes involves a bit of set theory: under ZFC plus the existence of a strong inaccessible cardinal $\kappa$, the foundational assumption of MacLane in Categories for the Working Mathematician, call a category moderate if its set of morphisms has size $\kappa$. For example, $Set$ is moderate.
(Street)
A moderate locally small category $C$ is a Grothendieck topos if and only if the Yoneda embedding
has a left exact left adjoint.
These results emphasize frames/toposes as algebras, where the morphisms are left exact left adjoints. The right adjoints to such morphisms are called geometric morphisms, and emphasize locales/toposes as spaces. This analogy, which plays an important but mostly tacit role in Joyal and Tierney, can be developed further along the following lines.
The frame of opens $O(X)$ corresponding to a locale $X$ is naturally a site:
Given a locale $X$, with frame of opens $O(X)$, say that a family of morphisms $\{U_i \to U\}$ in $O(X)$ is a cover if $U$ is the join of the $U_i$:
See for instance (MacLaneMoerdijk, section 5).
For $X$ a locale, write
for the sheaf topos over the category $O(X)$ equipped with the above canonical structure of a site.
Write Topos for the category of Grothendieck toposes and geometric morphisms.
This construction defines a full and faithful functor $Sh(-) :$ Locale $\to$ Topos.
This appears for instance as MacLaneMoerdijk, section IX.5 prop 2.
A topos in the image of $Sh(-)\colon Loc \to Topos$ is called a localic topos.
The functor $Sh(-)\colon Loc \to Topos$ has a left adjoint
given by sending a topos $\mathcal{E}$ to the locale that is formally dual to the frame of subobjects of the terminal object of $\mathcal{E}$:
This appears for instance as MacLaneMoerdijk, section IX.5 prop 3.
The functor $L$ here is also called localic reflection.
In summary this means that locales form a reflective subcategory of Topos:
In fact this is even a genuine full sub-2-category:
For all $X,Y \in$ Locale the 2-functor $Sh\colon$ Locale $\to$ Topos induces an equivalence of categories
This appears as (Johnstone, prop. C1.4.5).
The poset of subobjects $Sub_{\mathca{E}}(*)$ of the terminal object of $\mathcal{E}$ is equivalent to the full subcategory $\tau_{\leq -1}(\mathcal{E})$ of $\mathcal{E}$ on the $(-1)$-truncated objects of $E$:
A (-1)-truncated sheaf $X$ is one whose values over any object are either the singleton set, or the empty set
A monomorphism of sheaves is a natural transformation that is degreewise a monomorphism of sets (an injection). Therefore the subobjects of the terminal sheaf (that assigns the singleton set to every object) are precisely the sheaves of this form.
We may think of a frame as a Grothendieck (0,1)-topos. Then localic reflection is reflection of Grothendieck toposes onto $(0,1)$-toposes and is given by $(-1)$-truncation: for $X$ a locale, $Sh(X)$ the corresponding localic topos and $\mathcal{E}$ any Grothendieck topos we have a natural equivalence
which is
This is a small part of a pattern in higher topos theory, described at n-localic (∞,1)-topos.
The notion of locale may be identified with that of a Grothendieck (0,1)-topos. See Heyting algebra for more on this.
An ionad is to a topological space as a Grothendieck topos is to a locale.
A group object internal to locales or an internal groupoid in locales is a localic group or localic groupoid, respectively.
There is also a notion of internal locale, see also internal site.
An introduction to and survey of the use of locales instead of topological spaces (“pointless topology”) is in
This is, in its own words, to be read as the trailer for the book
that develops, among other things, much of standard topology entirely with the notion of locale used in place of that of topological spaces. See Stone Spaces for details.
Other dedicated textbooks are
and
where the latter develops topology from the point of view of its internal logic.
Plenty of material is also in part C (volume 2) of
Locales are also discussed in section IX.1 of
Lex totality is the subject of an article of Street,
The connection between locales and toposes via lex totality plays a tacit role throughout the influential monograph