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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 23rd 2019

Created the page.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJul 23rd 2019

Changed the wording slightly so that “element of a locale” refers to element of the corresponding frame.

I seem to have seen this language use recently. Personally I think it might be less confusing to say “opens of a locale” instead of “elements”.

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeJul 23rd 2019

In older books and papers (and, perhaps, some recent work sticking to older terminology) one often finds the word “locale” used to mean what we nowadays call “frame”, so that an element of the locale is the same as an element of the frame. But I would argue that the modern perspective, whereby a locale means an object of $Frm^{op}$ instead, means that “element of a locale” doesn’t really mean anything, since $Frm^{op}$ doesn’t have a canonical forgetful functor to $Set$. “Opens of a locale” seems a better term to me.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJul 23rd 2019

Change to use “open of a locale” and write $O(L)$ for the frame of opens of a locale $L$.

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 25th 2019
• (edited Jul 25th 2019)

since Frm^op doesn’t have a canonical forgetful functor to Set

Actually, Frm^op does have a canonical forgetful functor to Set: just take the right adjoint map of posets, which is guaranteed to exist.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJul 25th 2019

I suppose you could call that canonical, but it doesn’t have most of the usual properties of forgetful functors. Or, put differently, defining “a locale is a complete lattice satisfying the infinite distributive law of finite meets over arbitrary joins” and “a locale morphism is a monotone map preserving arbitrary meets whose left adjoint also preserves finite meets” is a pretty ad hoc definition of the category $Loc$, and doesn’t to my mind justify calling the elements of a frame “elements” of its corresponding locale.

1. By analogy we should also refer to a “sheaf of a Grothendieck topos” rather than an “object of a Grothendieck topos”. But I suspect that too many people identify toposes with their category of sheaves for that change of nomenclature to stick.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJul 27th 2019

Well, we don’t say “element of a Grothendieck topos”. A topos has both “points” and “objects”, generalizing how a locale has “points” and “opens”.