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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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, ...
Jakob Werner's user avatar
2 votes
0 answers
50 views

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 ...
Gro-Tsen's user avatar
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3 votes
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 (“...
Gro-Tsen's user avatar
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4 votes
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74 views

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 ...
Gro-Tsen's user avatar
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10 votes
4 answers
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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&...
Adrien Zabat's user avatar
9 votes
1 answer
484 views

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 ...
Emily's user avatar
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2 votes
0 answers
151 views

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 ...
Gro-Tsen's user avatar
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7 votes
1 answer
289 views

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 ...
Madeleine Birchfield's user avatar
4 votes
3 answers
363 views

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; ...
Madeleine Birchfield's user avatar
3 votes
1 answer
123 views

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 ...
Valery Isaev's user avatar
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0 answers
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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 \...
Hans's user avatar
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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. ...
user1892304's user avatar
4 votes
1 answer
168 views

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). ...
Simon Henry's user avatar
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7 votes
2 answers
668 views

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 ...
user477332's user avatar
5 votes
0 answers
235 views

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 ...
Cayley-Hamilton's user avatar
8 votes
1 answer
506 views

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 $U \cap V = \varnothing$,...
Arshak Aivazian's user avatar
6 votes
1 answer
197 views

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 $\...
user978360's user avatar
2 votes
0 answers
109 views

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 ...
BlackBoxedConchqueror's user avatar
5 votes
0 answers
155 views

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 $f: \mathbb R \to \mathbb R$ ...
wlad's user avatar
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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{...
user333306's user avatar
9 votes
2 answers
412 views

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 $\neg p\lor\neg\neg p$ ...
მამუკა ჯიბლაძე's user avatar
2 votes
1 answer
72 views

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 $x_n$ ...
Valery Isaev's user avatar
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5 votes
1 answer
220 views

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 ...
Bumblebee's user avatar
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46 votes
4 answers
5k views

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 ...
user877505's user avatar
8 votes
1 answer
321 views

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 ...
seldon's user avatar
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17 votes
1 answer
1k views

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 ...
user avatar
17 votes
1 answer
458 views

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 (...
Toby Bartels's user avatar
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4 votes
1 answer
188 views

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 $\mathbf{Frm}$, ...
Math Student 020's user avatar
7 votes
1 answer
918 views

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. ...
Paul Taylor's user avatar
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14 votes
1 answer
528 views

"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 ...
Simon Henry's user avatar
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16 votes
3 answers
868 views

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 ...
Mozibur Ullah's user avatar
6 votes
0 answers
146 views

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 ...
Angel Zaldívar's user avatar
25 votes
2 answers
1k views

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 ...
David Spivak's user avatar
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15 votes
5 answers
1k views

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 ...
Maxime Ramzi's user avatar
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4 votes
0 answers
101 views

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 ...
Math Student 020's user avatar
12 votes
3 answers
759 views

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 $\...
namsap's user avatar
  • 335
13 votes
2 answers
505 views

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] \...
Valery Isaev's user avatar
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9 votes
1 answer
472 views

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 ...
Helveticat's user avatar
43 votes
1 answer
4k views

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 ...
ಠ_ಠ's user avatar
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16 votes
2 answers
609 views

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 ...
Valery Isaev's user avatar
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12 votes
0 answers
397 views

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 $\mathrm{Set}[\mathbb{T}]$ ...
Ingo Blechschmidt's user avatar
5 votes
0 answers
250 views

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 ...
Ben's user avatar
  • 283
5 votes
1 answer
186 views

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 $...
Mike Shulman's user avatar
5 votes
0 answers
357 views

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 ...
Ben's user avatar
  • 283
5 votes
2 answers
193 views

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 ...
Ben's user avatar
  • 283
19 votes
2 answers
2k views

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, ...
HeinrichD's user avatar
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2 votes
0 answers
361 views

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-...
Rafał Gruszczyński's user avatar
5 votes
1 answer
126 views

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 ...
Simon Henry's user avatar
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3 votes
0 answers
418 views

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 ...
Angel Zaldívar's user avatar
5 votes
1 answer
304 views

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 ...
twocubes's user avatar
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