**1**

vote

**0**answers

206 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 ...

**3**

votes

**1**answer

52 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 ...

**4**

votes

**0**answers

113 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

**1**answer

260 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

**0**answers

76 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

**1**answer

128 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

**1**answer

112 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**

votes

**1**answer

75 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

**1**answer

117 views

### Is every frame monomorphism regular?

Is every monomorphism in $\mathbf{Frm}$, the category of frames, regular?

**5**

votes

**1**answer

139 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

**1**answer

146 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.

**4**

votes

**2**answers

265 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 $\mathcal{A}$ ...

**13**

votes

**1**answer

471 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 ...

**5**

votes

**0**answers

126 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 ...

**2**

votes

**1**answer

106 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

**1**answer

313 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

**2**answers

344 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

**0**answers

128 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 ...

**6**

votes

**1**answer

291 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

**3**answers

467 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

**2**answers

496 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

**0**answers

145 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

**1**answer

207 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 ...

**2**

votes

**1**answer

108 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

**1**answer

373 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

**1**answer

303 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

**1**answer

148 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 $ ...

**7**

votes

**2**answers

356 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

**0**answers

113 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

**1**answer

434 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 ...

**21**

votes

**0**answers

729 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 ...

**16**

votes

**0**answers

735 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

**3**answers

734 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 ...

**23**

votes

**3**answers

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

**1**answer

363 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

**5**answers

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 ...

**0**

votes

**1**answer

185 views

### Strong monics in the category of locales

Are there non-regular strong monics in the category of locales?

**5**

votes

**2**answers

508 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 ...