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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2011

    created focal point

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 7th 2011

    Interesting – I was expecting to see something about optics from a mathematical physics perspective. :-) Does anyone else besides Freyd-Scedrov use this terminology? (Although it doesn’t seem so bad – it’s almost opposite to “generic point”, i.e., a maximally smeared-out point.)

    Added a few examples to focal point.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2011

    Right. This was a link long time ago demanded by local topos. I simply wanted to un-gray that link. Eventually we should add a diambiguation and some physics there!

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 7th 2011

    Incidentally, is the Freyd cover of a Grothendieck topos EE a local topos over SetSet? It sort of feels like it wants to be, if we intuitively think of the Freyd cover as the result of adjoining a generic focal point to the “space” EE, as it were.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2011

    Isn’t this what Mike pointed out? Attributing it to C3.6.5 of the Elephant.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 7th 2011

    Okay, thanks Urs. I haven’t been following a lot of these discussions at all closely, because I’ve largely been away from mathematics period. (And, you and Mike have been covering a lot of ground recently.)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2011

    I have added a brief remark to this extent here. Maybe the more precise reference to Johnstone is: lemma C3.6.4.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeDec 7th 2011

    You have a great intuition, Todd! I don’t know that I would have come up with that immediately after seeing the definition of focal point.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 7th 2011

    Thanks, Urs!

  1. Added to focal point the example of the prime spectrum of a local ring and the locale theoretical characterisation.