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.

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Is pointless topology a generalization of point-set topology?

I heard about the pointless topology and so the theory of frames and locales. I’d would like to know how pointless topology is a generalization of point-set topology ? Is it a good thing to think of ...
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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 ...
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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},\...
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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 ...
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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 ...
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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.
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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. ...
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6 votes
2 answers
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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 ...
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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 ...
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7 votes
1 answer
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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' ...
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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 ...
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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, ...
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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 $\...
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2 answers
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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)=\...
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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 $r : L \to \kappa L$ ...
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5 votes
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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 ...
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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 ...
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1 vote
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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)...
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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.
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8 votes
1 answer
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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 ...
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1 vote
1 answer
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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.
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2 votes
1 answer
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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 ...
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3 votes
1 answer
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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 ...
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3 votes
1 answer
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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 ...
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4 votes
1 answer
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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 ...
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5 votes
1 answer
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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, ...
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3 votes
1 answer
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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 ...
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0 votes
1 answer
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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 ...
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3 votes
1 answer
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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 ...
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8 votes
1 answer
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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 ...
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5 votes
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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: ...
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10 votes
1 answer
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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 ...
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7 votes
2 answers
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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 ...
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