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.
83
questions
7
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1
answer
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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, ...
- 91
2
votes
0
answers
59
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 ...
- 28.7k
3
votes
1
answer
196
views
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 (“...
- 28.7k
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&...
- 201
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1
answer
486
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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 ...
- 9,727
2
votes
0
answers
152
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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 ...
- 28.7k
7
votes
1
answer
295
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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 ...
- 2,016
4
votes
3
answers
378
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; ...
- 2,016
3
votes
1
answer
129
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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 ...
- 4,340
0
votes
0
answers
114
views
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
$$
\mathcal{Sup} \leftrightarrows \...
- 127
2
votes
0
answers
93
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Concrete topological objects and notions in the category of locales
I have read Peter Johnstone's “The Point of Pointless Topology” and the idea that topological spaces are not quite the right abstraction for topology seems, at least philosophically, rather appealing. ...
- 121
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votes
1
answer
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The locale of morphisms vs a morphism to an ultrapower?
I'm fixing some type of structure $\Sigma$ (possibly multi-sorted, with functions symbols and relation symbols, though assuming it single sorted with only relation symbols wouldn't change anything). ...
- 39.4k
7
votes
2
answers
674
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Every Grothendieck topos can be built from localic topoi
Theorem 2 in these notes[1] states that, roughly, that each Grothendieck topos can be built (using limits and colimits) from localic topoi. To what extent is that related to the theorem of Joyal and ...
- 73
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votes
0
answers
235
views
Is this property of continuous maps equivalent to some more familiar condition?
Let $f : X \rightarrow Y$ be a continuous map. Suppose that, for each collection of open sets $\{ V_i \}_{i \in I}\subset X $,
$$ \bigcup_{U \subset Y \text{ open}, \ f^{-1}(U) \subset \bigcup_{i \in ...
- 4,647
8
votes
1
answer
509
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 $U \cap V = \varnothing$,...
- 6,164
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votes
1
answer
198
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 $\...
- 63
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votes
0
answers
109
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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 ...
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 $f: \mathbb R \to \mathbb R$ ...
- 4,752
7
votes
1
answer
439
views
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{...
- 375
9
votes
2
answers
413
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$ ...
- 17.3k
2
votes
1
answer
74
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$ ...
- 4,340
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votes
1
answer
226
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 ...
- 1,007
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 ...
- 571
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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 ...
- 1,023
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 ...
user147820
17
votes
1
answer
461
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 (...
- 2,584
5
votes
1
answer
212
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}$, ...
- 201
7
votes
1
answer
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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.
...
- 7,756
14
votes
1
answer
531
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 ...
- 39.4k
16
votes
3
answers
872
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 ...
- 2,057
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 ...
- 159
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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 ...
- 8,327
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votes
5
answers
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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 ...
- 12.6k
4
votes
0
answers
101
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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 ...
- 201
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votes
3
answers
768
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 $\...
- 335
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votes
2
answers
508
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] \...
- 4,340
9
votes
1
answer
475
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 ...
- 93
43
votes
1
answer
4k
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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 ...
- 5,877
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votes
2
answers
613
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 ...
- 4,340
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votes
0
answers
399
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}]$ ...
- 3,102
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 ...
- 293
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votes
1
answer
188
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 $...
- 64.1k
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votes
0
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366
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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 ...
- 293
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 ...
- 293
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, ...
- 5,382
2
votes
0
answers
361
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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-...
- 1,092
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votes
1
answer
127
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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 ...
- 39.4k
3
votes
0
answers
420
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 ...
- 159
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 ...
- 125
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votes
0
answers
101
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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 ...
- 39.4k