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Questions taking place in the category of locales, which is given by the opposite of the category of frames. Also appropriate for questions about pointless topology.

55 votes
Accepted

Constructive algebraic geometry

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. …
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25 votes
0 answers
1k 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). …
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23 votes
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Locales as geometric objects

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). …
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17 votes
4 answers
2k 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 …
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14 votes
1 answer
565 views

"Scott completion" of dcpo

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 …
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14 votes

Localic or topos-theoretic definition of $\operatorname{Spec}$

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 …
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13 votes
Accepted

Locales in constructive mathematics

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. …
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13 votes
Accepted

What are projective locales / injective frames?

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 $B_\kappa$. …
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12 votes

Locales as spaces of ideal/imaginary points

And it makes the connection with those "locales of imaginary points" very clear. It should be equivalent to the classical description.. … The first step is to look at the "space of all thing", i.e. the classifying locales of the theory of "thing". So "thing" has to be a nice (geometric) notion so that such a classyfing space exists. …
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12 votes
1 answer
529 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 in …
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12 votes
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What's the localic reflection of a presheaf topos?

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 …
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11 votes
Accepted

Which topological manifolds do not correspond to strongly Hausdorff locales?

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. …
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10 votes

What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

Of course because we are working with locales one cannot necessarily "construct" an object at the end, because a non-empty locale does not neccessary have a points. … I like to sum up this by the fact that in locales theory "every discrete space is (geometrically) countable", but this more of a slogan. …
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8 votes
0 answers
103 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'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 …
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6 votes
0 answers
254 views

Are regular epi of locale stably epic?

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. …
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