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.
59
questions
14
votes
1answer
515 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 ...
13
votes
1answer
223 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 (...
3
votes
1answer
93 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 $\mathbf{Frm}$, ...
7
votes
1answer
352 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.
...
12
votes
1answer
340 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 ...
13
votes
3answers
630 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 ...
6
votes
0answers
121 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 ...
24
votes
2answers
731 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 ...
12
votes
5answers
732 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 ...
4
votes
0answers
91 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 ...
12
votes
3answers
610 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 $\...
13
votes
2answers
374 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] \...
9
votes
1answer
406 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 ...
33
votes
1answer
3k 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 ...
13
votes
2answers
405 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 ...
9
votes
0answers
219 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 $\mathrm{Set}[\mathbb{T}]$ ...
5
votes
0answers
234 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 ...
5
votes
1answer
155 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 $...
4
votes
0answers
157 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 ...
5
votes
2answers
150 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 ...
14
votes
2answers
889 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, ...
2
votes
0answers
332 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-...
4
votes
1answer
87 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 ...
3
votes
0answers
305 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 ...
5
votes
1answer
285 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 ...
7
votes
0answers
94 views
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
1answer
164 views
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 ...
2
votes
1answer
203 views
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 ...
1
vote
1answer
100 views
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
votes
1answer
165 views
Is every frame monomorphism regular?
Is every monomorphism in $\mathbf{Frm}$, the category of frames, regular?
7
votes
1answer
232 views
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 ...
3
votes
1answer
176 views
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.
13
votes
3answers
1k views
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....
14
votes
1answer
840 views
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 ...
6
votes
0answers
180 views
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
1answer
161 views
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
votes
1answer
446 views
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 ...
4
votes
2answers
405 views
German translation of “locale” (from pointless topology)
Is there an established German translation of "locale"? The term appears mostly untranslated as "Locale"; a single time I've seen "Lokal". Where I'm located, we say "Örtlichkeit" or "Ort".
Quoting ...
3
votes
0answers
177 views
pullback of a morphism of locale which is an isomorphism?
Let $A,B$ be two locales over a locale $X$, and $f:A\rightarrow B$ a morphism of locale over $X$. Let also $g:X'\rightarrow X$ be a surjection of locale such that the pullback of $f$ along $g$ is an ...
10
votes
1answer
501 views
What's the link between topological spaces as locales and topological spaces as infinity-groupoids?
I've seen texts that talk about topological spaces being essentially locales, like Topology via Logic by Vickers, and texts related to homotopy theory that talk about topological spaces being ...
14
votes
4answers
1k views
reference request : constructive measure theory
As the title said, I would like to know if constructive measure theory has been developed somewhere ?
I am more precisely interested in the (constructive) theory of completely continuous valuation on ...
5
votes
2answers
941 views
Embedding a brouwerian lattice into a boolean lattice
I have already asked a similar question at
https://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra
but have received no answer.
Sorry, I ask a ...
0
votes
0answers
216 views
Is there a translation invariant measure on an infinite dimensional space 'without points'?
This is just a reference request. I thought I'd come across a paper demonstrating that there is a translation invariant measure on an infinite-dimensional space without 'points' whilst browsing the ...
1
vote
1answer
257 views
Strongly Hausdorff(Isbell Hausdorff)
Can I get a finite Isbell Hausdorff frame which has no Isbell Hausdorff subframe? If not possible is there one such in infinite case?
See the book [1] for definitions.
Picado, Jorge, and Ales Pultr. ...
4
votes
1answer
156 views
On the openness of the map X^I -> X * X.
Hello !
Let $X$ be a locale or a topological spaces. $I$ denote the unit interval of the real numbers, and $X^I$ the space of function from $I$ to $X$ (The locale exponential if $X$ is a locale or ...
12
votes
1answer
470 views
Are $\infty$-topoi determined by their localic points ?
Hello !
If $T$ is an infinity topos, then you can consider the infinity category of geometric morphism from $Sh_{\infty}(\mathcal{L})$ to $T$ for any locale $\mathcal{L}$. This associate to $T$ an ...
11
votes
2answers
390 views
Defining measures over frames in place of $\sigma$-algebras
Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, ...
1
vote
1answer
182 views
Intersection of open sublocale of a compact regular locale ?
Hello !
It's well know that any sublocale of regular locale is the intersection of a familly of open sublocale. Hence if $X$ is a regular locale, the map which to a sublocal $Y \subset X$ associate $ ...
9
votes
2answers
482 views
Given a Grothendieck topos, what does its localic groupoid look like? [duplicate]
Possible Duplicate:
Toposes (topoi) as classifying toposes of groupoids
For example, if a topos E is the object classifier, or the preseaf topos on a small category C, is there a way of ...
2
votes
0answers
128 views
surjection of localic infinity toposes?
Hello!
Is there a simple 'topological' condition to detect whenever a morphism of locales $f : X \rightarrow Y$ induces a surjection of infinity-toposes $f : \mathrm{Sh}_{\infty}(X) \rightarrow \...
