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Results tagged with locales
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user 11640
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.
9
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1
answer
366
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Is an open map with open relative diagonal necessarily a local homeomorphism?
Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta_f : X \to X \times_Y X$ is an open embedding of locales. … An open embedding of locales is a morphism $f : X \to Y$ that is isomorphic to the inclusion of some open sublocale of $Y$. …
- 15.9k
12
votes
Every Grothendieck topos can be built from localic topoi
They are (it is?) the same theorem, but emphasising different aspects.
We can exploit the object classifier to get from the formulation in terms of (pseudo)colimits to the "elementary" formulation in …
- 15.9k
12
votes
Localic or topos-theoretic definition of $\operatorname{Spec}$
This is ultimately the same construction as the one Simon Henry describes, but you might like the different perspective.
Definition.
Let $A$ be a commutative rig and let $L$ be a distributive lattice. …
- 15.9k
14
votes
Accepted
The real numbers object in Sh(Top)
Following a suggestion of Thomas Holder, we can close the gap as follows:
For each object $Y$ in $\mathbf{T}$, there is a pseudonatural local geometric morphism $\mathbf{Sh}(\mathbf{T}_{/ Y}) \to \m …
- 15.9k
7
votes
1
answer
125
views
Universally closed implies proper for locales
What about locales?
Is there an intuitionistic proof?
It seems to me possible to improve the proof mentioned above, for topological spaces at least. …
- 15.9k
2
votes
Universally closed implies proper for locales
It turns out that Vermeulen has essentially answered the question in [A note on stably closed maps of locales].
The argument there implies:
Theorem.
Let $g : X \to S$ be a morphism of locales. …
- 15.9k