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stub at locally compact locale
Added to locally compact locale a class of examples of locally compact locales which may constructively fail to be spatial, namely spectra of rings.
In the process I noticed that our entry on the spectrum of a commutative ring needs some love. For instance, it is missing a constructively sensible construction (as a locale or directly as a topos, as referenced in this recent math.SE question), the formulation of its universal property, and a discussion of its relation to Hakim’s very general spectrum construction and to the relative spectrum construction commonly used in algebraic geometry. When time allows, I’ll have a go at these additions.
I have added the references, like this:
* Christopher F. Townsend: *A categorical proof of the equivalence of local compactness and exponentiability in locale theory*, [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]], **47** 3 (2006) 233-239 [[numdam:CTGDC_2006__47_3_233_0](http://www.numdam.org/item/CTGDC_2006__47_3_233_0)]
* [[J. Martin E. Hyland]]: *Function spaces in the category of locales* in: *Continuous Lattices*, Lecture Notes in Mathematics **871**, Springer (1981) [[doi:10.1007/BFb0089910](https://doi.org/10.1007/BFb0089910)]
Is there a reference for locally compact locales (in the sense of continuous frames) all being spatial?
Re #5: The article spatial locale has a reference.
Thanks, and thank you Urs
https://terrytao.wordpress.com/2011/05/24/locally-compact-topological-vector-spaces/
Can someone help me to resolve my confusion about locally compact topological ℝ-vector spaces?
According to the two facts we’ve established above, along with “all locally compact Hausdorff topological ℝ-vector spaces are finite dimensional”, I am struggling to understand [G,ℝ] for a compact hausdorff topological group G. Does this example imply that [G,-] does not send locally compact objects to locally compact objects?
What does [G,R] denote? I guess you realize that the image of a map G→R is bounded, so that any such homomorphism must be trivial.
Re #9: I believe [−,−] denotes the internal hom in the category of locally compact locales. I do not think there was any intention to take homomorphisms of localic groups G→R, but merely morphisms of locales.
Okay, thanks. I would say that exponentiability of an object G does not imply exponentiability of [G,R], even if R is exponentiable. A simple example is that the discrete space ℕ is exponentiable, but ℕℕ (Baire space of sequences) is not locally compact.
Thanks very much, this answers what I meant to ask.
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