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    • CommentRowNumber1.
    • CommentAuthortyler bryson
    • CommentTimeAug 9th 2016
    The present nLab page for Filter ( claims the following:

    "If L is a complete join-semilattice, then Filters(L) is a complete lattice."

    I don't see how this is true and no reference is given. Can someone shed light on this? After two days with it, I suspect this is a typo...but am not sure.

    Much obliged,
    Tyler Bryson
    • CommentRowNumber2.
    • CommentAuthorDexter Chua
    • CommentTimeAug 9th 2016

    A complete join-semilattice is automatically a complete meet-semilattice, so by the first point in the list, Filters(L)Filters(L) is a complete join-semillatice, hence a complete lattice.

    • CommentRowNumber3.
    • CommentAuthortyler bryson
    • CommentTimeAug 9th 2016

    Thank you. I should have seen that. I do think the wording on that bullet point could be improved. As it stands, I expect I am not the only person who’ll incorrectly infer that the completeness of the lattice Filters(L)Filters(L) is a consequence of passing to filters and not an easy byproduct of the order on LL. I may submit an edit.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeAug 27th 2016

    I think that the point of those bullet points is supposed to be about how f *f_* has various nice properties when ff does. So the first sentence of each is sort of just introductory: if LL has some structure, then Filters(L)Filters(L) has some structure, OK, but now here's what's interesting: if ff preserves the structure on LL, then f *f_* preserves the structure on Filters(L)Filters(L), and that's what we want to talk about! At least, that's the attitude, I think.