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  1. 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 jj. Provisionally I named it “bb”, in allusion to the flat modality. I refrained from naming it “\flat”, 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.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 11th 2016

    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.

    • CommentRowNumber3.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeMay 11th 2016
    • (edited May 11th 2016)

    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 X jXX_j \hookrightarrow X is essential from the internal point of view of the ambient sheaf topos Sh(X)Sh(X) 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.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 11th 2016

    I’m all for collecting hard-to-access category theory papers ;-)

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2016

    This reminds me to add a pointer to Lima’s talk to the nnLab here

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 11th 2016

    It’s mildly interesting that a sublocale X jXX_j \hookrightarrow X is essential from the internal point of view of the ambient sheaf topos Sh(X)Sh(X) if and only if it is open (instead of essential).

    That sounds like a locale version of the fact that a geometric morphism FEF\to E is essential from the internal point of view of EE 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.

  2. I thought so too. But, unless I have made a calculational mistake, being an essential sublocal is a local property: A sublocale X jXX_j \hookrightarrow X is essential if and only if, for an open covering X= iU iX = \bigcup_i U_i, the sublocales X jU iU iX_j \cap U_i \hookrightarrow U_i 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.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMay 11th 2016

    Well, just being local is not sufficient to be internally characterizable.