The locales tag has no wiki summary.
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1answer
71 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.
3
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2answers
119 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 $\mathcal{A}$ ...
12
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1answer
300 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 ...
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0answers
65 views
Are regular epi of locale stably epic?
It is well know that the category of locale is not a regular category, that is the pullback of a regular epimorphism is not always a regular epimorphism: for example, the classical counterexample ...
2
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1answer
93 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 ...
3
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1answer
191 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
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2answers
312 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 ...
2
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0answers
103 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 ...
5
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1answer
211 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 ...
8
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3answers
336 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 ...
2
votes
2answers
351 views
Embedding a brouwerian lattice into a boolean lattice
I have already asked a similar question at
http://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra
but have received no answer.
Sorry, I ask a ...
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0answers
140 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
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1answer
194 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 ...
1
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1answer
98 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 ...
10
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1answer
345 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
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1answer
283 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
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1answer
127 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 $ ...
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votes
2answers
306 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
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0answers
110 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 ...
1
vote
1answer
372 views
Counterexemple to Urysohn's lemma in a topos without denombrable choice ?
Hello !
The Urysohn's Lemma assert that in every topological spaces which is normal two closed subset may be separated by a real valued function. It's proof use axiom of countable choice (but not the ...
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0answers
642 views
$\infty$-topos and localic $\infty$-groupoids?
It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales).
For the record, this is proved by, starting ...
13
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0answers
538 views
Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?
Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra.
Moreover, a morphism of commutative von Neumann algebras induces
a continuous morphism of the corresponding ...
8
votes
3answers
651 views
Localic locales? Towards very pointless spaces by iterated internalization.
One can think of locales as (generalizations of) topological spaces which don't necessary have (enough) points. Of course when one studies locales, one "actually" studies frames,
certain sorts of ...
19
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3answers
2k views
Locales and Topology.
As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...
11
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1answer
350 views
Do strict pro-sets embed in locales?
It is well-known that the category of profinite groups (by which I mean Pro(FiniteGroups), i.e. the category of formal cofiltered limits of finite groups) is equivalent to a full subcategory of ...
4
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5answers
1k views
Stone Spaces, Locales, and Topoi for the (relative) beginner
I am currently reading Vickers' text "topology via logic" and Peter Johnstone's "stone spaces", and I understand the material in both of these texts to pertain directly to constructions in elementary ...
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1answer
179 views
Strong monics in the category of locales
Are there non-regular strong monics in the category of locales?
5
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2answers
458 views
Definition of Category of Locales
In the wikipedia entry for 'frames and locales', pains are taken to distinguish between the category of locales - defined to be the opposite of the category of frames - and the category whose objects ...
