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I created a stub about essential sublocales. I’ll polish the entry a bit more in a few hours and then link to it from other entries.
I’m not sure how to name the left adjoint to the nucleus . Provisionally I named it “”, in allusion to the flat modality. I refrained from naming it “”, since this symbool seems most often to refer to the induced action on subobjects or types.
Unfortunately I don’t have access to Kelly and Lawvere’s article On the complete lattice of essential localizations. It probably contains a few more properties of essential sublocales which I’d like to copy to the nLab entry.
Wow that article doesn’t seem easy to come by. My library has old issues of the Belg.Math.Soc journal in storage, I’m sure I could get a scan of the article in question.
Thanks to Guilherme Frederico Lima (who, incidentally, gave a nice talk titled From Essential Inclusions to Local Geometric Morphisms at Topos à l’IHÉS), I now have a copy of that article (and will gladly send it to anyone who asks). :-) I didn’t get around to announce that yesterday. Thanks for your offer to scan it!
For the moment, I’m done with editing the entry.
It’s mildly interesting that a sublocale is essential from the internal point of view of the ambient sheaf topos if and only if it is open (instead of essential). At the moment I don’t know an internal characterization of essentiality, but I’ll look for it.
I’m all for collecting hard-to-access category theory papers ;-)
This reminds me to add a pointer to Lima’s talk to the Lab here
It’s mildly interesting that a sublocale is essential from the internal point of view of the ambient sheaf topos if and only if it is open (instead of essential).
That sounds like a locale version of the fact that a geometric morphism is essential from the internal point of view of iff it is locally connected. I kind of doubt there is a characterization of essentiality that is internal to the codomain; as I’ve understood it the whole point is that locally connected / open is the “internal version” of essentiality, by making the left adjoint indexed over the base.
I thought so too. But, unless I have made a calculational mistake, being an essential sublocal is a local property: A sublocale is essential if and only if, for an open covering , the sublocales are essential. Since the internal language cannot characterize properties which are not local, but being essential is local, there is some hope that it could be internalized.
Well, just being local is not sufficient to be internally characterizable.
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