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1. • CommentRowNumber1.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMay 9th 2016
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexI created a stub about [[essential sublocale|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 $j$. Provisionally I named it "$b$", 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.

   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 $j$. Provisionally I named it “$b$”, 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.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 11th 2016
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexWow 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMay 11th 2016
   • (edited May 11th 2016)
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexThanks to Guilherme Frederico Lima (who, incidentally, gave a [nice talk](https://www.youtube.com/watch?v=YsoGN91Rh_s) 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 $X_j \hookrightarrow X$ is essential from the internal point of view of the ambient sheaf topos $Sh(X)$ 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.

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

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 11th 2016
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   Author: DavidRoberts
   Format: MarkdownItexI'm all for collecting hard-to-access category theory papers ;-)

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

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis reminds me to add a pointer to Lima's talk to the $n$Lab [here](https://ncatlab.org/nlab/show/essential+geometric+morphism#Lima15)

   This reminds me to add a pointer to Lima’s talk to the $n$Lab here

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMay 11th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> It's mildly interesting that a sublocale $X_j \hookrightarrow X$ is essential from the internal point of view of the ambient sheaf topos $Sh(X)$ if and only if it is *open* (instead of essential). That sounds like a locale version of the fact that a geometric morphism $F\to E$ is essential from the internal point of view of $E$ 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.

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

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

7. • CommentRowNumber7.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMay 11th 2016
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexI thought so too. But, unless I have made a calculational mistake, being an essential sublocal _is_ a local property: A sublocale $X_j \hookrightarrow X$ is essential if and only if, for an open covering $X = \bigcup_i U_i$, the sublocales $X_j \cap U_i \hookrightarrow U_i$ 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.

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

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeMay 11th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWell, just being local is not *sufficient* to be internally characterizable.

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