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2
votes
1answer
73 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
votes
1answer
162 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
288 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 ...
1
vote
0answers
90 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
votes
1answer
190 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
votes
3answers
287 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
279 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 ...
0
votes
0answers
137 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
186 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
vote
1answer
82 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
votes
1answer
322 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
1answer
273 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
107 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 $ ...
6
votes
2answers
282 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
105 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
353 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 ...
17
votes
0answers
615 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
votes
0answers
503 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
610 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 ...
18
votes
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
votes
1answer
340 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
votes
5answers
937 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 ...
0
votes
1answer
173 views

Strong monics in the category of locales

Are there non-regular strong monics in the category of locales?
5
votes
2answers
436 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 ...