# 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.

84
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### G-topological spaces and locales

Consider the following generalization of topological spaces:
Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...

2
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### What is known about sublocales defined by regular nuclei?

(For basic terminology, which is supposed to be standard anyway, see this other question, which inspired this one.)
I am interested in nuclei $j\colon L\to L$ on a frame $L$ which are regular elements ...

3
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1
answer

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### Computing the Heyting operation on the frame of nuclei

(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound (“...

4
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### Constructively valid reference for the soberness of discrete spaces and points of a locale coproduct

I am looking for constructively valid references for the following two related facts:
discrete topological spaces are sober,
the points of a locale coproduct are the disjoint union of the points of ...

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4
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### Localic or topos-theoretic definition of $\operatorname{Spec}$

Usually, the construction of the spectrum of a commutative ring starts with defining the points of $\operatorname{Spec}(A)$, and constructing a topology with the closed sets being the "zeroes&...

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### Is there a good theory of 2-locales?

Topological spaces and locales are two closely related notions meant to capture the concept of an abstract space, the latter of which admits a certain "noncommutative" generalisation known ...

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### Concrete description of “DeMorganian” open sets

Let me begin with a few definitions. My question will be basically how to simplify them to something more manageable. The motivation for these definitions is given at the end.
Let $X$ be a ...

7
votes

1
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### Status of the fundamental theorem of algebra for the locale of real numbers

In constructive mathematics without any choice at all, it is well known that the fundamental theorem of algebra cannot be proven for the Dedekind real numbers. On the other hand, Wim Ruitenberg showed ...

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### The field structure on the locale of real numbers

It is well known how to derive the field operations from the construction of the real numbers as the Dedekind completion of the rational numbers and as the Cauchy completion of the rational numbers; ...

3
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1
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### Localic maps given by series

Maps between real numbers are often defined by convergent series. For example, to define the exponential map, we can just prove that series
$$
\sum_{n = 0}^{\infty} \frac{x^n}{n!}
$$
converges, which ...

0
votes

0
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### Colimits in the category of suplattices

I want to compute coequalizers in the category $\mathcal{S}up$ of complete lattices and $\bigvee$-preserving maps. One way (I think?) is to use the dual equivalence
$$
\mathcal{Sup} \leftrightarrows \...

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### Concrete topological objects and notions in the category of locales

I have read Peter Johnstone's “The Point of Pointless Topology” and the idea that topological spaces are not quite the right abstraction for topology seems, at least philosophically, rather appealing. ...

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### The locale of morphisms vs a morphism to an ultrapower?

I'm fixing some type of structure $\Sigma$ (possibly multi-sorted, with functions symbols and relation symbols, though assuming it single sorted with only relation symbols wouldn't change anything). ...

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### Every Grothendieck topos can be built from localic topoi

Theorem 2 in these notes[1] states that, roughly, that each Grothendieck topos can be built (using limits and colimits) from localic topoi. To what extent is that related to the theorem of Joyal and ...

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### Is this property of continuous maps equivalent to some more familiar condition?

Let $f : X \rightarrow Y$ be a continuous map. Suppose that, for each collection of open sets $\{ V_i \}_{i \in I}\subset X $,
$$ \bigcup_{U \subset Y \text{ open}, \ f^{-1}(U) \subset \bigcup_{i \in ...

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### What is the status of Jordan's theorem in constructive mathematics in the language of locales?

By constructive mathematics in this matter we mean intuitionistic ZF (*).
In the language of locales, the Jordan curve can be defined as $f\colon S^1 \to X$ such that "if $U \cap V = \varnothing$,...

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### Connections between $0$-toposes and $1$-toposes (Grothendieck and elementary)

This is a crosspost from math.stackexchange.
A Grothendieck $0$-topos is the same as a frame (see here). Another relationship between Grothendieck toposes and frames is the following: whenever $\...

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### Explicit description of the canonical $\pi_G: \mathrm{Sh}(G_0)\to B_S(\mathbf{G})$

Given a localic groupoid $\mathbf{G} = (G_1\overset{d_0}{\underset{d_1}{\rightrightarrows}}G_0)$ and letting $B\mathbf{G}$ denote its classifying topos, I'm looking for a explicit description of the ...

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### Within pointless topology inside of choiceless constructivism, prove that division is possible

In Frank Waaldijk's paper on the foundations of constructive analysis, Waaldijk shows that various definitions of "continuous function" for functions of the form $f: \mathbb R \to \mathbb R$ ...

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### Does the functor $\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$ have an adjoint?

Consider the category $\mathbf{Top}$ of topological spaces, the category $\mathbf{Topos}$ of toposes and geometric morphisms, and the category $\mathbf{Loc}$ of locales. Let
$$\mathrm{Sh}\colon\mathbf{...

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2
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### What are projective locales / injective frames?

Judging by the compact regular case, and more generally the spatial case, regular projectivity of locales, resp. regular injectivity of frames, must have something to do with $\neg p\lor\neg\neg p$ ...

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votes

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### Convergence of localic maps

We can define a limit of a sequence of points in a locale in the usual way: $x$ is a limit of $\{ x_i \}_{i \in \mathbb{N}}$ if, for every open $U$ containing $x$, there exists $N$ such that $x_n$ ...

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### What locales correspond to Manifolds?

I am studying the categorical equivalence between (sober) topological spaces and (spatial) locales with enough points. As the title implies, I am interested in finding localic analogues of both ...

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### How to rewrite mathematics constructively?

Many mathematical subfields often use the axiom of choice and proofs by contradiction. I heard from people supporting constructive mathematics that often one can rewrite the definitions and theorems ...

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### What's the localic reflection of a presheaf topos?

$\newcommand{\Psh}{\operatorname{Psh}}
\newcommand{\Sh}{\operatorname{Sh}}
\newcommand{\O}{{\mathcal{O}}}$
Let $X$ be a locale, $\O(X)$ the corresponding frame.
What's the localic reflection of $\Psh ...

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1
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### Best introductory texts on pointless topology

As I understand it, there are three canonical textbooks on pointless topology: the classic "Stone Spaces" by Johnstone, "Topology via Logic" by Steve Vickers, and the newer "Frames and Locales" by ...

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### Combination topological space and locale?

The traditional theory of topological spaces (as formalized by Bourbaki) starts with a set of points, then builds a structure on that. In contrast, the theory of locales starts with a frame of opens (...

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1
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### Product of topological spaces and product of corresponding locales

Let $\Omega : \mathbf{Top} \to \mathbf{Loc}$ the functor that takes a topological space to its locale of opens.
For a locale morphism $f$, write $f^*$ for the correspoding morphism in $\mathbf{Frm}$, ...

7
votes

1
answer

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### Differentiability of the distance function from a (variable) point to a (fixed) set

The distance of from a point $x$ to a set $A$ is defined by
$$ d(x,S) = \inf\{d(x,a)\mid a\in A\}, $$
where you may choose the setting to be $\mathbb R^n$,
a Banach space or a complete metric space.
...

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### "Scott completion" of dcpo

If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup_I a_i \in U $ for ...

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### What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales:
...the single most important fact which distinguishes locales from spaces: the ...

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### Spatiality of products of locally compact locales

In Johnstone´s Sketches of an Elephant Volume 2, page 716,
lemma 4.1.8 states that for spatial locales $X$ and $Y$ with $X$ locally compact then the locale product $X\times Y$ is spatial.
Is this ...

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### Another notion of exactness: how to refine it, and where does it fit?

There are many notions of “exactness” in category theory, algebraic geometry, etc. Here I offer another that generalizes the category of frames, the notion of valuation (from probability theory), and ...

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### Locales as spaces of ideal/imaginary points

I posted this question on MSE a few days ago, but got no response (despite a bounty). I hope it will get more answers here, but I'm afraid it might not be appropriate as I'm not sure it's actually ...

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### Compact subspace of sober space

We know from lemma 1.2.5 in part C of Sketches of an Elephant (by Johnstone) that both open and closed subspaces of a sober space are again sober. This raises the following question.
Question: Is a ...

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### Is it possible to completely embed complete Heyting Algebras into upsets of a poset?

Let $H$ be a Heyting algebra. It is a well-known result that there is a partially ordered set (Kripke frame) X such that there is an embedding of Heyting algebras $f: H \to \mathsf{Up}(X)$, where $\...

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### Constructive proofs of existence in analysis using locales

There are several basic theorems in analysis asserting the existence of a point in some space such as the following results:
The intermediate value theorem: for every continuous function $f : [0,1] \...

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### Which topological manifolds do not correspond to strongly Hausdorff locales?

I'm toying with the idea of using locales as a way to define topological manifolds without beginning with points, largely for philosophical reasons.
In this context I think I want to redefine a ...

43
votes

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### Constructive algebraic geometry

I was just watching Andrej Bauer's lecture Five Stages of Accepting Constructive Mathematics, and he mentioned that in the constructive setting we cannot guarantee that every ideal is contained in a ...

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### Locales in constructive mathematics

It is well known that locales are much more well behaved in a constructive setting than topological spaces. Nevertheless, many authors develop the theory of locales in classical mathematics. Are there ...

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### What does the localic reflection of a classifying topos classify?

Let $\mathbb{T}$ be a geometric theory. Let $\mathrm{Set}[\mathbb{T}]$ be its classifying topos, such that geometric morphisms from any (cocomplete) topos $\mathcal{E}$ into $\mathrm{Set}[\mathbb{T}]$ ...

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### Uniqueness of localic analogue of Radon-Nikodym derivatives

In classical probability theory, Radon-Nikodym derivatives are unique up to measure-0 sets of the background measure. I am wondering whether the following statement, intended to state an analogue of ...

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1
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### Topological regularity for toposes

A topological space $X$ is regular if a point not belonging to a closed set can be separated from it by disjoint opens. Equivalently, this means if $x\in U$ for an open set $U$, then there are opens $...

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### Inductive generation of non-spatial locales

Is there an example of a locale/formal space which is not spatial (i.e., one can prove it is not spatial, rather than something like $\mathbb{R}$, where spatiality is independent in a constructive ...

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### Products of double-negation sublocales (and probability distributions on them)

In locale theory, one can produce from any locale $A$ its double negation sublocale $A_{\neg\neg}$ via the nucleus which maps an open $U$ of $A$ to $\neg \neg U$, which is the interior of the closure ...

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### Locales as geometric objects

There is the following analogy:
$$\begin{array}{cc} \text{frames} & - & \text{commutative rings} \\ | && | \\\text{locales} & - & \text{affines schemes}\end{array}$$
Here, ...

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0
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### The theory of frames and locales as elementary topology [closed]

In What is elementary geometry? (pdf) Alfred Tarski defined elementary geometry to be
that part of Euclidean geometry which can be formulated and established without the help of any set-...

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1
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### Relative local compactness for locales?

I am looking for informations on the relative version of local compactness for locales:
If $f:X \rightarrow Y$ is a morphism of locales I want to say that $f$ is relatively locally compact if ...

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### Topos Theory, internal Heyting Algebra

Given a topos $\mathcal{E}$ with subobject classifier $\Omega$.
If we denote by $N\Omega$ the former of all local operators on $\Omega$, that is, Lawvere–Tierney topologies of $\mathcal{E}$, it is ...

5
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1
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### Not sure how to fix an error in the Handbook of Categorical Algebra (vol 3)

In the proof of Theorem 1.5.7 (in which it is shown the the nuclei on a locale, when ordered by pointwise partial order, themselves form a locale) in the computation at the bottom of p.35, there is ...