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(for example the diagonal of any non discrete finite 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 internally … It is equivalent to the fact that $X$ is exponentiable in the category $Locale/Y$ of locales over $Y$. …

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 $ …

Let $A,B$ be two locales over a locale $X$, and $f:A\rightarrow B$ a morphism of locale over $X$. …

Let $X$ be a locale or a topological space. $I$ denote the unit interval of the real numbers, and $X^I$ the space of functions from $I$ to $X$ (The locale exponential if $X$ is a locale or the set of …

There is a pretty good notion of convergence of maps of locales, though I have never seen anything in the literature about it (maybe I should write something about it ?). …

on the nlab for cat/top/poset also work for locales. … epimorphism followed by a monomorphism and that every strong epimorphism of locales is a regular epimorphism. …

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've also put the tag general topology because if we restrict to spatial locales then we have a question of purely about point set topology and sheaves over (sober) topological space which is probably …

Thus the inclusion of locales $X \hookrightarrow Y$ is given by the quotient of frames $\mathcal{O}(Y) \to \mathcal{O}(X)$ defined as $V \mapsto \bigvee V$ and whose right adjoint is $v \mapsto {\downarrow …

Here is a fairly general methods for this sort of thing : Step 1) We give a constructive proof that for each (Dedekind) real $x$, the serie $\sum \frac{x^n}{n!}$ converge. We define $exp(x)$ as the l …

As I said any locally compact Hausdorff topological space is a strongly hausdroff locally compact locales. … But having only studied the constructive theory of locales I know very little of paracompactness in this framework so I leave to someone else to comment or answer about this. …

The answer is yes. It appears for example as lemma C3.1.15 in Johnstone's sketches of an elephant. Roughly, it can be proved by working in the internal logic of the target (hence assuming that the t …

For your first question, if $X$ and $Y$ are two boolean locale then $X \times Y$ is boolean only if $X$ or $Y$ is discrete. So unless $\neg \neg A$ or $\neg \neg B$ are discrete, $\neg \neg A \times \ …

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 \mathrm{Sh}_{ … It's not enough to assume that f is a surjection of locales: indeed, if we take a topological space $X$ such that $\mathrm{Sh}_{\infty}(X)$ is not hypercomplete, and $X^{\mathrm{disc}}$ is its space of …

answer your more specific questions: 1) One generally says that a locale is discrete if its diagonal map is an open embeddings and if the map $X \rightarrow 1$ is open (in the sense of open morphisms of locales … But there is no other implications: boolean locales are Hausdorff but not spatial nor discrete, and spatial locale can be both discrete and non discrete and both Hausdorff and non Hausdorff. …