# Questions tagged [locales]

For questions about locales, a generalization of topological spaces which need not have points. Their study is called pointless topology.

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### What is the universal property of the prime spectrum of a commutative rig?

Let $A$ be a commutative rig, i.e. a commutative monoid equipped with a unital associative commutative bilinear multiplication and let $L$ be a distributive lattice. For the purposes of this question, ...
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### How to represent a collection of closed sets in a locale of opens?

Given a collection of closed sets $\mathcal{F}$ of a topological space $X$, how do I relate this with a locale $\mathcal{O}(X)$ of opens of $X$?. The obvious case is when we have a collection of ...
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### Stone-Čech compactification for locales

I'm reading about the Stone-Čech compactification for locales, which (I think) states that for every locale $L$ there exists a compact regular locale $\kappa L$ and a morphism $r : L \to \kappa L$ ...
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### What are are some illustrative (non-)examples of proper morphisms?

"Proper" is an adjective used to describe a morphism of spaces—topological spaces, schemes, locales, etc—that is sufficiently nice and has some neat properties. Between topological spaces a morphism ...
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### Rings and locales

Let $A$ be a ring. Given a monoid (=multiplicative system) $S \subseteq A$ one can define the localization of $A$ at $S$ as the $A$-algebra $$j_S \colon A \to A[S^{-1}]$$ which is universal among the ...
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### Locales as spaces of ideal/imaginary points

I recently saw a video of a presentation of Andrej Bauer here about constructive mathematics; and there are two examples of locales he mentions that strike me : he explains quickly what the space of ...