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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 …

I'm fixing some type of structure $\Sigma$ (possibly multi-sorted, with functions symbols and relation symbols, though assuming it single sorted with only relation symbols wouldn't change anything). L …

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). …

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 …

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 …

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 $Y$. …

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 …

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. …

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 $X$. …

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 in …

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 …

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 …