Questions tagged [locales]
Questions taking place in the category of locales, which is given by the opposite of the category of frames. Also appropriate for questions about pointless topology.
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Constructive status of the meta-theory of intuitionistic propositional logic
Intuitionistic propositional logic has several kinds of models. Bezhanishvili and Holliday [1] showed that these models form a neat hierarchy:
Kripke
Beth
Topological
Dragalin
Heyting
in the order ...
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In a topos, the subobject classifier is a compact locale?
Constructively, it is true that for any distributive lattice $D$, its ideal completion $Idl(D)$ is a coherent locale (thus in particular compact). It is also well-known that $\Omega$ in a topos $\...
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Does $\neg(j^{**}(\bot))=\neg\neg(j^{*}(\bot))$ hold for all nuclei $j$ and nucleic/localic pseudocomplements $*$/$\neg$?
The nucleus $j \rightarrow k$ in the frame $N(A)$ of nuclei on a locale $A$ is defined on page 52 of Johnstone's Stone Spaces as the meet $$(j\rightarrow k)(a):=\bigwedge_{b\geq a}(j(b)\rightarrow k(b)...
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When are the points of a topological space complemented as sublocales?
The $T_D$ separation condition for topological spaces has various formulations. In a stub on the nLab, $T_D$ is defined to mean that every point is "open in its closure". In Picado and Pultr'...
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Is there a purely localic description of the real line?
In this math overflow question about characterising the real line purely topologically, the accepted answer is that it is a connected and locally connected regular topological space that is seperable ...
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In what sense is the double negation topology functorial?
Given any locale $L$, one can form the double negation sublocale $L_{\neg \neg} \hookrightarrow L$ from the nucleus $\neg \neg$. In what sense is this construction functorial? Or asked slightly ...
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Proof that the Vietoris locale VX is compact if the locale X is compact
Let $X$ be a locale[+]. Then there is the Vietoris locale construction, $V$, introduced by Johnstone (it's in his Stone Spaces, but see also the 1985 paper, 'Vietoris Locales and Localic Semilattices')...
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Dimensionality of the rational numbers locale
In topological spaces, a well-known result is that $\mathbb{Q}^n$ are homeomorphic for all positive integers $n$. This means there are no topological notion of dimensionality for "rational spaces&...
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Sober topological spaces
The sober topological spaces seem to me very nice becouse they are basically just locales. What is the analogue in condensed mathematics?
The functor from topological spaces to the full pro etale site ...
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Is there any work on point-free topological vector spaces?
Is there any substantial work on topological vector spaces in the point-free setting? It seems like this might be able to clarify some of the traditional pathologies of the subject.
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A lattice/topos-theoretic construction of the Boolean algebra of measurable subsets modulo nullsets
Suppose we take the viewpoint that the right category for measure theory is the category of measurable locales or equivalently, the opposite category of the category of commutative Von Neumann ...
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Sublocales corresponding to filters
It is a well known fact that the sublocales of the locale $L$ are defined by idempoten $\wedge$-semilattice endomorphisms, known as nuclei. Each nucleus $j$ of a locale $L$ also defines a filter $\...
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Is the inclusion of an étale complete localic groupoid $\mathbb{G}$ in its localic category $\gamma \mathbb{G}$ dense?
This is a question about categories internal to the category of locales, $\mathbf{Loc}$.
An étale complete localic groupoid $\mathbb{G}$ gives rise to a localic category, $\gamma \mathbb{G}$, by ...
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Preimage of a sublocale by a morphism of locales: description by nucleus?
For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs ...
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How algebraic can the dual of a topological category be?
(I'm going to try to use definitions from Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich, and Strecker, since both of the adjectives in the title of my question seem to have at ...
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