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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There are more sublocales than subspaces.
In this article: https://arxiv.org/pdf/2406.12486v1 it is written:
Given a topological space $X$, there are typically more
sublocales in its frame of opens $\Omega(X)$ than subspaces in X.
Can you ...
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Are localic maps monotonic?
A frame homomorphism $f:L\to M$ is increasing which means that if $l_1 \le l_2$ then $f(l_1)\le f(l_2)$.
What about the corresponding localic map $\tilde f :M\to L$ is it increasing?
All what we know ...
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Monomorphisms in $Loc$ need not be one-to-one
Let $Loc$ be the category of locales, why a monomorphsim in $Loc$ need not be one-to-one? I know how to prove that a monomorphism is one-to-one in the category Set but I don't know why the same proof ...
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The functor $Sob \to Loc $ is faithful
I want to show that the functor $O: Sob \to Loc$ from the sober topological spaces to the locales is faithful. So take $X,Y$ two sober topological spaces. I want to show that the map
$$O_{X,Y}: Hom(X,...
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Localic slices of an étendue
An étendue is a Grothendieck topos $\mathcal{E}$ containing a well-supported object $A$ such that $\mathcal{E}/A$ is localic. This does not imply that all slices are localic (for example, the topos of ...
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sigma-algebra vs sigma-frame
A $\sigma$-algebra $\mathcal{A}$ on a set $X$ is a family of subsets of $X$ such that:
$\phi\in \mathcal{A}$.
$\mathcal{A}$ is closed under countable unions.
$\mathcal{A}$ is closed under ...
- 1,068
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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....
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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 ...
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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 ...
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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 ...
- 538
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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 ...
- 538
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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 ...
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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 ...
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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:
...
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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 ...
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