Questions tagged [locales]

For questions about locales, a generalization of topological spaces which need not have points. Their study is called pointless or point-free topology. They are related with lattice-theoretic structures such as frames, Heyting algebras, Boolean algebras as well as topos theory. Use in conjunction with those tags as necessary.

Filter by
Sorted by
Tagged with
2 votes
1 answer
31 views

Congruence lattice of a semiring

A famous result of Funayama and Nakayama states that the congruence lattice of any lattice is a distributive lattice [1]. Also, it can be proved that the lattice is a frame/ complete Heyting algebra 2....
MathCosmo's user avatar
  • 846
0 votes
0 answers
36 views

Zariski opens of an affine space and (po)sites

There is a very neat presentation of the Zariski open sets in $\operatorname{Spec}(R)$: it is freely generated (under arbitrary unions and finite intersections, which distribute over each other) by ...
Trebor's user avatar
  • 4,575
2 votes
1 answer
72 views

The equivalences between points in a locale in constructive mathematics

I am currently following the definitions of the book Frames and Locales and those in the articles Pointfree topology and constructive mathematics and Topo-logie. There are at least three ways of ...
Dylan Facio's user avatar
0 votes
0 answers
52 views

Names of definitions in frames/locales/topologies

Background definitions. A frame $(\mathbb X,\leq)$ is a partially ordered set with arbitrary joins (least upper bounds) and finite meets (greatest lower bounds), such that finite meets distribute ...
Jim's user avatar
  • 518
1 vote
1 answer
120 views

Condition for maximal filter of a lattice to be completely prime

Background definitions. A frame $(\mathbb X,\leq)$ is a partially ordered set with arbitrary joins (least upper bounds) and finite meets (greatest lower bounds). A filter $F\subseteq \mathbb X$ is a ...
Jim's user avatar
  • 518
3 votes
1 answer
99 views

Abstract characterisation of spatial frames / spatial locales

By definition, a spatial locale is a locale that is isomorphic to the locale of open sets of a topological space. See for example here: https://ncatlab.org/nlab/show/spatial+locale Question: Is there ...
Jim's user avatar
  • 518
2 votes
1 answer
59 views

Morphisms of frames induce morphisms of sites

One can associate a site with an arbitrary frame by defining coverages with suprema. According to Johnstone (Sketches of an Elephant, 2.3.20), "If $A$ and $B$ are frames, made into sites via ...
Daniel Rogozin's user avatar
1 vote
1 answer
54 views

Internal homs in a locale seems to be symmetric?

TL;DR: just read the last sentence of the question. I'm watching a video series on intro level topos theory. It introduces sheaves on a locale as part of the motivation. It claims the following fact: ...
ice1000's user avatar
  • 262
1 vote
2 answers
124 views

Products of spaces of covering dimension zero

Recall that a space has covering dimension zero if every open cover of it can be refined to a disjoint open cover. Question 1. Does the product of two spaces of covering dimension zero have covering ...
Zhen Lin's user avatar
  • 90.2k
-2 votes
1 answer
115 views

Advantages of locale theory in calculus [closed]

I'm considering whether to learn point-set topology or pointless topology. Are there advantages to using locales in calculus?
Hayatsu's user avatar
  • 363
0 votes
0 answers
35 views

What is this separation property of locales called

Trying to come up with a point-free formulation of the $T_1$ axiom I thought of the following condition on a frame/locale $L$: $$\forall U \in L \, . \, U \wedge \bigwedge \{ V \in L : U \vee V = \top ...
Jonas Frey's user avatar
2 votes
2 answers
166 views

Getting Stone duality from the duality between sober spaces and spatial frames

Here it says: Probably the most general duality that is classically referred to as "Stone duality" is the duality between the category Sob of sober spaces with continuous functions and the ...
user1005113's user avatar
1 vote
0 answers
93 views

Characterization of stalk functors on category of sheaves over a space, and over a locale

Let $X$ be a topological space (somewhat abusively also let it denote the category of open sets in $X$) and $\mathsf{Set}_X$ the category of $\mathsf{Set}$-valued sheaves over $X$. Let $\mathcal{F},\...
P-addict's user avatar
  • 1,502
1 vote
1 answer
58 views

Does the category of $C^*$-algebras embed into the category of frames?

The category of commutative $C^*$-algebras is dual to the category of LCH topological spaces. My understanding is that authors in operator algebras often understand a 'noncommutative LCH space' to be ...
namsos's user avatar
  • 1,022
2 votes
0 answers
97 views

Do open continuous maps/local homeomorphisms between locales possess adjoints?

Recently I started learning "theory of locales" (point-free topology) by my-self. While being a very beautiful, natural subject and parallel to point-set topology, some of its notions are ...
Bumblebee's user avatar
  • 18.3k
2 votes
1 answer
66 views

Cofree frame given a suplattice

Does the forgetful functor from frames to suplattices have a right adjoint? Does it have a left adjoint? I am thinking it has a right adjoint, and that there is a "cofree" frame.
user avatar
2 votes
1 answer
241 views

Pull back of sheaves has left adjoint if and only if the morphism between topological spaces is a local homeomorphism [closed]

I am looking for a way to show that $f:X\rightarrow Y$ is a local homeomorphism of topological spaces if and only if $f^* :\operatorname{Sh}(Y)\rightarrow \operatorname{Sh}(X)$ has a left adjoint. ...
user373827's user avatar
7 votes
2 answers
194 views

Open maps of spaces vs open maps of locales

A continuous function between topological spaces $f:X\to Y$ is called open, if $f[U]\in\mathcal{O}(Y)$ for all $U\in\mathcal{O}(X)$. A morphism of locales $f:X\to Y$ is called open, if the associated ...
Jonas Frey's user avatar
3 votes
0 answers
86 views

Duality Between Semilattices and Totally Disconnected locally Compact Hausdorff Spaces

On page 18 of this paper, the author states that there is a duality (correspondence?) between semilattices (i.e., abelian semigroups of idempotents) and totally disconnected locally compact Hausdorff ...
user193319's user avatar
  • 7,940
10 votes
1 answer
788 views

Why is a geometric theory called “geometric”?

In topos theory, the notion of “being geometric” often comes up. Some examples are: geometric morphisms, geometric logic, and geometric theories. For instance, here's a quote from Steve Vickers' ...
affibern's user avatar
  • 433
1 vote
1 answer
57 views

Proving that the pointful and the pointless forms of the well-inside relation coincide

In Stone Spaces by Johnstone, a pointless version of the well-inside order is given: Definition 1. Let $L$ be a locale and let $x, y \in L$. $x$ is said to be well-inside $y$ iff \begin{equation*} x ...
affibern's user avatar
  • 433
9 votes
0 answers
239 views

What is the universal property of the prime spectrum of a commutative rig?

Let $A$ be a commutative rig, i.e. a commutative monoid equipped with a unital associative commutative bilinear multiplication and let $L$ be a distributive lattice. For the purposes of this question, ...
Zhen Lin's user avatar
  • 90.2k
1 vote
0 answers
72 views

Necessary and sufficient conditions for a canonical mapping $\sigma$ to be surjective in an $L$-set.

A locale is a complete lattice $L$ such that: $\bullet$ $a\wedge\bigvee_{b\in B}b=\bigvee_{b\in B}(a\wedge b)$ for all $a\in L$ and $B\subseteq L$. An $L$-set is a set $X$ together with a function $\...
Daniel Kawai's user avatar
  • 1,005
2 votes
2 answers
104 views

Are constant functions continuous in constructive mathematics?

The standard proof that a constant function $c: X \to Y$, $x \mapsto y_0$ is continuous proceeds as follows: if $U \subseteq Y$ is open, then either $c^{-1}(U)=X$ if $y_0 \in U$, or $c^{-1}(U)=\...
ಠ_ಠ's user avatar
  • 10.7k
1 vote
1 answer
158 views

Stone-Čech compactification for locales

I'm reading about the Stone-Čech compactification for locales, which (I think) states that for every locale $L$ there exists a compact regular locale $\kappa L$ and a morphism $r : L \to \kappa L$ ...
Math Student 020's user avatar
5 votes
0 answers
254 views

What are are some illustrative (non-)examples of proper morphisms?

"Proper" is an adjective used to describe a morphism of spaces—topological spaces, schemes, locales, etc—that is sufficiently nice and has some neat properties. Between topological spaces a morphism ...
Mike Pierce's user avatar
  • 18.9k
3 votes
1 answer
162 views

Rings and locales

Let $A$ be a ring. Given a monoid (=multiplicative system) $S \subseteq A$ one can define the localization of $A$ at $S$ as the $A$-algebra $$j_S \colon A \to A[S^{-1}]$$ which is universal among the ...
user avatar
1 vote
0 answers
50 views

About a proof that localic maps are open iff their left adjoints are complete Heyting algebra homomorphisms

I'm reading the proof of the statement in the title in Picado and Pultr's book Frames and Locales (Proposition 7.2). The authors first obtain the formula $x\wedge \phi(a) = y\wedge \phi(a)\iff f^*(x)...
greens's user avatar
  • 53
0 votes
1 answer
120 views

Regular locale is spatial

Guided by topology I'm expecting this to be true, but I cannot find a proof for the following statement: Every regular locale is spatial.
Math Student 020's user avatar
9 votes
2 answers
510 views

Do products preserve colimits in the category of locales?

Does the functor $X\times-:\mathbf{Loc}\to\mathbf{Loc}$ preserve small colimits for all locales $X$? The reason that I'm interested in this question is that the same property fails in the category of ...
Oscar Cunningham's user avatar
2 votes
1 answer
408 views

Some examples of non-spatial frames.

I'm looking for some examples of non-spatial frames. (a frame is non-spatial iff not isomorphic to any frames have forms of topologies for some sets) A simpler example is better for me.
Sho Banno's user avatar
6 votes
1 answer
313 views

Does the frame of open sets in a topological space or locale really have all meets?

According to the nLab article on locales, a frame has all meets by the adjoint functor theorem: This seems a bit strange to me, since it's well-known that an infinite intersection of open subsets is ...
ಠ_ಠ's user avatar
  • 10.7k
3 votes
1 answer
295 views

Is there a classifying topos for locales?

Is there a Grothendieck topos $F$ such that, for any Grothendieck topos $E$, the category of geometric morphisms $$E\rightarrow F$$ is equivalent to the category of locales internal to $F$? I suspect ...
Oscar Cunningham's user avatar
5 votes
1 answer
98 views

If $f$ is an epimorphism of frames, it is surjective as function? [closed]

If $f$ is an epimorphism of frames, then is it surjective as a function?. The frames behaves so much like topological spaces (moreover the locales which are the opposite category), so the question is ...
Math.mx's user avatar
  • 1,939
4 votes
1 answer
221 views

Free frame generated by a poset

Suppose I have a poset $P$, is there a "best" frame for $P$; that is a frame $L$ with a monotone map $P\to L$ that is universal ? What if I add some nice conditions on $P$: the $P$'s I'm interested ...
Maxime Ramzi's user avatar
  • 43.6k
7 votes
1 answer
335 views

The etale locale of a sheaf?

It's well-known that sheaves over a topological space are equivalent to etale spaces over the same space. Now if we replace "topological space" by "locale", we can still define sheaves over a locale, ...
Maxime Ramzi's user avatar
  • 43.6k
3 votes
1 answer
371 views

Are frames (the lattice kind) complete?

There seems to be conflicting information about frames and complete Heyting algebras. Everyone seems to agree on the fact that frames are lattices in which any subset has a supremum, but not every ...
E Hekkelman's user avatar
0 votes
1 answer
71 views

Confused about definition: "Pointwise Equalizer"

I am reading these notes on topos theory, and I have a small confusion about Proposition 1.16 on page 12. What is the difference between a "pointwise equalizer" ($K$ in proposition 1.16) and the ...
Aurel's user avatar
  • 633
4 votes
1 answer
294 views

Locales as spaces of ideal/imaginary points

I recently saw a video of a presentation of Andrej Bauer here about constructive mathematics; and there are two examples of locales he mentions that strike me : he explains quickly what the space of ...
Maxime Ramzi's user avatar
  • 43.6k
8 votes
1 answer
440 views

Algebraic topology on locales

My question is essentially in the title: is there a well-developped theory of algebraic pointless topology, that is algebraic topology on locales ? If not, would it make sense, i.e. would it be ...
Maxime Ramzi's user avatar
  • 43.6k
5 votes
0 answers
76 views

Localic group as generalizations of topological groups

I have read about what locales and frames are (basic objects in pointfree topology) and now I'be seen that there exists localic groups as a generalization of topological groups. I have some questions: ...
kryomaxim's user avatar
  • 2,882
10 votes
1 answer
262 views

Two questions on completely regular filters in locales

I'm reading the exposition of the Stone-Čech compactification for locales in Johnstone's book Stone Spaces. In Chapter IV Paragraph 2.2, Johnstone constructs the Stone-Čech compactification of a ...
Alex Kruckman's user avatar
8 votes
2 answers
999 views

Info on the locale of surjections from the Natural Numbers to the Real Numbers

On the nlab page for locales, it states that there is locale for the surjections from the Naturals to the Reals. This locale has no points (i.e. elements), since there are no such surjections, but the ...
Christopher King's user avatar