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Jean-Baptiste Vienney (Jun 02 2024 at 16:30):I read on the nlab that the category of all locales is not cartesian closed (in this entry: nice category of spaces). Is there any category of locales which is cartesian closed as we have categories of topological spaces which are cartesian closed?
Fernando Yamauti (Jun 02 2024 at 16:56):https://ncatlab.org/nlab/show/continuous+poset ?
Jean-Baptiste Vienney (Jun 02 2024 at 18:15):Thank you!
Jean-Baptiste Vienney (Jun 02 2024 at 18:27):In this entry, there is written:
I think there is a mistake in the parenthesis after "locally compact" and we should write "(i.e. every point of the sobrification has a neighbourhood basis of compact sets)", right?
Graham Manuell (Jun 02 2024 at 19:05):Yes, I think that is what was meant, but I agree it could have been clearer.
Graham Manuell (Jun 02 2024 at 19:10):You probably should specify some additional conditions, as otherwise I could say something like finite locales or discrete locales, though I don't know any cartesian closed subcategories of locales that contain for instance.
You might also be interested in the paper "A Cartesian closed extension of the category of locales" by Reinhold Heckmann.
Graham Manuell (Jun 02 2024 at 19:12):On the other hand, in this paper me and @Peter Faul discuss whether you should expect locales to have exponentials.
Jean-Baptiste Vienney (Jun 02 2024 at 19:20):Graham Manuell said:
You probably should specify some additional conditions, as otherwise I could say something like finite locales or discrete locales, though I don't know any cartesian closed subcategories of locales that contain for instance.
But is locally compact, so its lattice of open sets is a continuous lattice, no?
Jean-Baptiste Vienney (Jun 02 2024 at 19:25):By the way, the cartesian closed category of locales of @Fernando Yamauti is the opposite of the category of continuous lattices and not posets, right?
Graham Manuell (Jun 02 2024 at 20:01):is locally compact and thus exponentiable. This does not mean it lies in a cartesian closed category. I'm pretty sure is not locally compact.
Also, continuous lattices form a cartesian closed category, but firstly these don't need to be distrubtive and secondly the morphisms are wrong, and in particular go in the wrong direction. They act like special spaces, not special frames.
Fernando Yamauti (Jun 03 2024 at 00:40):Jean-Baptiste Vienney said:
By the way, the cartesian closed category of locales of Fernando Yamauti is the opposite of the category of continuous lattices and not posets, right?
Locally compact locales are spatial (non-constructively). But weird stuff happens after exponentiation. For instance, the extended long ray has a non-spatial path space. Features like that allow it to be path connected in topos theory, but not path connected in ordinary topology (think as a generalised path linking two ordinary points).
Jonas Frey (Jun 03 2024 at 03:50):A standard example of a full subcategory of topological spaces that is cartesian closed is the category of [[compactly generated Hausdorff spaces]]. Since Hausdorff spaces are all sober, and sober spaces are equivalent to spatial locales, there is an equivalent full subcategory of locales.
Jonas Frey (Jun 03 2024 at 03:51):I'm not sure at the moment if compactly generated weak Hausdorff spaces are also sober, but I wouldn't be surprised if they are.
Fernando Yamauti (Jun 03 2024 at 05:16):Jonas Frey said:
A standard example of a full subcategory of topological spaces that is cartesian closed is the category of [[compactly generated Hausdorff spaces]]. Since Hausdorff spaces are all sober, and sober spaces are equivalent to spatial locales, there is an equivalent full subcategory of locales.
I think exponentiation is not quite the same in this case. The example I've mentioned are two compact Hausdorff spaces such that the exponential (of locales) is not spatial.
John Baez (Jun 03 2024 at 09:20):Graham Manuell said:
Yes, I think that is what was meant, but I agree it could have been clearer.
Please fix it, folks!
Graham Manuell (Jun 03 2024 at 11:18):Jonas Frey said:
A standard example of a full subcategory of topological spaces that is cartesian closed is the category of [[compactly generated Hausdorff spaces]]. Since Hausdorff spaces are all sober, and sober spaces are equivalent to spatial locales, there is an equivalent full subcategory of locales.
Ah, yes. This is true. I forgot about it since it hasn't been worked out constructively.
Todd Trimble (Jun 03 2024 at 20:55):Jean-Baptiste Vienney said:
In this entry, there is written:
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I think there is a mistake in the parenthesis after "locally compact" and we should write "(i.e. every point of the sobrification has a neighbourhood basis of compact sets)", right?
The standard and morally correct definition of locally compact space, in the absence of the Hausdorff condition, is exactly as you put it. This is stated in [[locally compact space]], although the first sentence is a somewhat goofball opener. I don't think there is any mistake.
A later comment that the morphisms are in the wrong direction is correct.
Jonas Frey (Jun 04 2024 at 00:34):Fernando Yamauti said:
Jonas Frey said:
A standard example of a full subcategory of topological spaces that is cartesian closed is the category of [[compactly generated Hausdorff spaces]]. Since Hausdorff spaces are all sober, and sober spaces are equivalent to spatial locales, there is an equivalent full subcategory of locales.
I think exponentiation is not quite the same in this case. The example I've mentioned are two compact Hausdorff spaces such that the exponential (of locales) is not spatial.
Yes, the exponentials in CGHaus generally don't coincide with any existing exponentials in Top, since the products in Top in CGHaus don't generally coincide either. This also true for variations such as compactly generated weak Hausdorff spaces, but doesn't seem to bother algebraic topologists.
Jonas Frey (Jun 04 2024 at 00:36):Ahh, now I see that that's not really what you were talking about. Yes, the example with the non-spatial exponential in locales is intriguing as well!
Morgan Rogers (he/him) (Jun 04 2024 at 07:32):Since locales are not cartesian closed, you should really expect exponentiation in any cartesian closed category of locales not to coincide with exponentiation in the category of locales, no?
Jonas Frey (Jun 04 2024 at 17:12):Morgan Rogers (he/him) said:
Since locales are not cartesian closed, you should really expect exponentiation in any cartesian closed category of locales not to coincide with exponentiation in the category of locales, no?
Well, there can still be cartesian closed subcategories whose exponentials are also exponentials in the ambient category. A not very interesting example is given by finite discrete spaces/locales.
Morgan Rogers (he/him) (Jun 04 2024 at 18:43):Sure, but if you have an object which is not exponentiable in Loc (or not repeatedly exponentiable) then any CCC containing it must have exponentials which can't coincide with those in Loc because the latter don't exist, and it should be considered a neat coincidence rather than an expectation if there is a coincidence on those that do exist.