# 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.

72
questions

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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). ...

6
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2
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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 ...

5
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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 ...

8
votes

1
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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 $U \cap V = \varnothing$,...

5
votes

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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 $\...

2
votes

0
answers

105
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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 $f: \mathbb R \to \mathbb R$ ...

7
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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{...

9
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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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### 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 $x_n$ ...

3
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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 ...

16
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1
answer

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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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393
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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 (...

4
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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

647
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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.
...

12
votes

1
answer

430
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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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0
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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 ...

25
votes

2
answers

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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 ...

4
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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] \...

9
votes

1
answer

444
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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 ...

40
votes

1
answer

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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 ...

16
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2
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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 ...

5
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1
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171
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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 ...

17
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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, ...

2
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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-...

4
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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 ...

3
votes

0
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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 ...

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### Locales satisfying DC?

Is there a nice (topological) characterization of the locales such that the axiom of dependant choices holds in the internal logic of the topos of sheaves ? I would also be interested in the case of ...

8
votes

1
answer

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### Detecting positive endomaps of the formal reals

A locale is a sort of "formal topological space", which "may not have enough points to separate its open sets". For instance, there is a "locale of all real numbers that are both rational and ...

3
votes

1
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### Exponential locales and a pointless version of the compact-open topology?

TL;DR: compact-open topology for Homs of locales?
Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales.
For two locales, $A$ and $B$, is there a nice way to make an ...

2
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1
answer

120
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### Lebesgue covering dimension for locales

A space $(X,\tau)$ is said to have Lebesgue covering dimension $\leq n$ (for some $n\in\mathbb{N}$) if every open covering $\cal U$ has a refinement $\cal V$ such that for every $x\in X$ the set ${\...

3
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1
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194
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### Is every frame monomorphism regular?

Is every monomorphism in $\mathbf{Frm}$, the category of frames, regular?

9
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1
answer

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### Is an open map with open relative diagonal necessarily a local homeomorphism?

Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta_f : X \to X \times_Y X$ is an open embedding of locales. Does it follow that $f : X \to Y$ is a ...

4
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1
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### Is there a straightforward way to define a differentiable structure on a localic manifold?

I'd ideally like a categorical definition of differentiability that can then be trivially translated into locales. Barring this, I'm still interested in whether the notion make sense for locales.

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3
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### About a construction of Borel $\sigma$-algebra associated to a lattice

Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$).
Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ i....

15
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1
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### The real numbers object in Sh(Top)

If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of ...

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### Are regular epi of locale stably epic?

It is well know that the category of locales is not a regular category, that is the pullback of a regular epimorphism is not always a regular epimorphism: for example, the classical counterexample ...

3
votes

1
answer

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### Disjoint arrows in the category of locales

Call two arrows $f$ and $g$ disjoint if the pullback of $f$ by $g$ is the initial object. Here's my question: Does there exist a sublocale $j: J\to L$ which is not disjoint with any other (non-...

4
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### Coequalizers in the category of algebras of the double power locale monad

$\mathbf{Loc}$ is the category of locales, and $\mathbb{P}$ is the double power locale monad on it. Consider the category $\mathbf{Loc}^{\mathbb{P}}$, of algebras of this monad. Does anyone know ...