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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Regular locale is spatial

Guided by topology I'm expecting this to be true, but I cannot find a proof for the following statement: Every regular locale is spatial.
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Do products preserve colimits in the category of locales?

Is the functor $X\times-:\mathbf{Loc}\to\mathbf{Loc}$ colimit preserving for all locales $X$? The reason that I'm interested in this question is that the same property fails in the category of ...
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28 views

Some examples of non-spatial frames.

I'm looking for some examples of non-spatial frames. (a frame is non-spatial iff not isomorphic to any frames have forms of topologies for some sets) A simpler example is better for me.
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A certain property of non-spatial frames

Let $L$ be a frame (complete distributive bounded lattice which satisfies: $a\wedge\bigvee_{i\in I}x_i=\bigvee_{i\in I}a\wedge x_i)$, and $\mathrm{pt}(L)$ be the set of all two-valued frame ...
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Does the frame of open sets in a topological space or locale really have all meets?

According to the nLab article on locales, a frame has all meets by the adjoint functor theorem: This seems a bit strange to me, since it's well-known that an infinite intersection of open subsets is ...
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112 views

Is there a classifying topos for locales?

Is there a Grothendieck topos $F$ such that, for any Grothendieck topos $E$, the category of geometric morphisms $$E\rightarrow F$$ is equivalent to the category of locales internal to $F$? I suspect ...
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1answer
38 views

If $f$ is an epimorphism of frames, it is surjective as function? [closed]

If $f$ is an epimorphism of frames, then it is surjective as function?. The frames behaves so much like topological spaces (moreover the locales which are the opposite category), so the question is ...
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1answer
95 views

Free frame generated by a poset

Suppose I have a poset $P$, is there a "best" frame for $P$; that is a frame $L$ with a monotone map $P\to L$ that is universal ? What if I add some nice conditions on $P$: the $P$'s I'm interested ...
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81 views

The etale locale of a sheaf?

It's well-known that sheaves over a topological space are equivalent to etale spaces over the same space. Now if we replace "topological space" by "locale", we can still define sheaves over a locale, ...
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1answer
64 views

Are frames (the lattice kind) complete?

There seems to be conflicting information about frames and complete Heyting algebras. Everyone seems to agree on the fact that frames are lattices in which any subset has a supremum, but not every ...
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1answer
40 views

Confused about definition: “Pointwise Equalizer”

I am reading these notes on topos theory, and I have a small confusion about Proposition 1.16 on page 12. What is the difference between a "pointwise equalizer" ($K$ in proposition 1.16) and the ...
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1answer
129 views

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 ...
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Algebraic topology on locales

My question is essentially in the title: is there a well-developped theory of algebraic pointless topology, that is algebraic topology on locales ? If not, would it make sense, i.e. would it be ...