Questions tagged [locales]

For questions about locales, a generalization of topological spaces which need not have points. Their study is called pointless topology.

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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, ...
6
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2answers
388 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 ...
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21 views

How to represent a collection of closed sets in a locale of opens?

Given a collection of closed sets $\mathcal{F}$ of a topological space $X$, how do I relate this with a locale $\mathcal{O}(X)$ of opens of $X$?. The obvious case is when we have a collection of ...
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54 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 $\...
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2answers
75 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)=\...
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1answer
35 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.
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1answer
29 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 $r : L \to \kappa L$ ...
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71 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 ...
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1answer
60 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 ...
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0answers
36 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)...
8
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1answer
177 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 ...
3
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1answer
60 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 ...
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1answer
81 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.
2
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1answer
68 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 ...
6
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1answer
122 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, ...
2
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1answer
149 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 ...
4
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1answer
121 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 ...
3
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1answer
168 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 ...
3
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1answer
96 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 ...
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1answer
51 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 ...
6
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1answer
164 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 ...