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aren't new, they are just bigger locales so there is no need for new objects (the missing corner is "locales" again). … map of locales from $X$ to $L$, so they are just locales with a specific set of points marked). …

Here is I think a counter-example to the precise proposed formula in the question. Take $X= \mathbb{Q}$ with the discrete topology, with $Y = \mathbb{R}$ withe its usual topoly and the map $f:X \to Y$ …

If $X$ is locally compactly generated then $X$ is compactly generated because every locally compact space is compactly generated. So given $f:X \to Y$ a map such that $f\circ i$ is continuous for ever …

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 …

The Zariski spectrum is essentially the classifying topos for prime ideal of $A$, or to be more precise, the classifying topos for subsets of $A$ that are "complement of prime ideals of $A$". The prec …

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 …

Alternatively, you can use the fact that these operations are well defined on Dedekind real numbers constructively and the methods explained here allow you to turn these into maps between the corresponding locales … Here again, to talk about inverse you have to restrict to Dedekind real that are $>0$ or $< 0$ which in terms of classifying locales corresponds to the open subspace $\mathbb{R} - \{0\}$ …

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). …

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 …

More precisely it induces an equivalence between "sober topological spaces" and "spatial locales", where spatial locales are the locales "having enough points" in a precise technical sense. … There are however some locales which have no points at all. …

As the question explicitely refers to "the language of locales" I will be (from now on) only refering to the localic version when talking about $\mathbb{R}^n$ and its subspace. … Now I'm completely convinced by the paper Andrej Bauer linked that the following is constructively* valid: Theorem : Let $j:S^1 \to \mathbb{R}^2$ be any monomorphisms of locales (between the formale locales …

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 \ …

So the short answer is that there is no non-empty projective locales for essentially any reasonable class of epimorphisms you can think of (except maybe proper maps). … Here is some clarification on the construction of locales $B_\kappa$. …

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 ?). …

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 …