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nLab > Latest Changes: interactions of images and pre-images with unions and intersections

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2017

    At interactions of images and pre-images with unions and intersections I have added pointer to Lawvere 69 and there at Adjointness in Foundations I added a bit more text and cross-references.

    • CommentRowNumber2.
    • CommentAuthorVitalyR
    • CommentTimeFeb 7th 2021
    I'm reading this article, and find a little contradiction between the text and formula in the properties 2. of Proposition 2.1.
    Should "contained in" be removed in the explanation of the properties 2. of Proposition 2.1?
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 8th 2021

    Let me see… It looks okay to me: The intersection of images may be larger than the image of the intersection.

    In an extreme case, take S1 and S2 to two disjoint copies of the same non-empty abstract set, and each with the same image under f. Then the image of the intersection is empty (because the intersection itself is), but the intersection of the images is the image of either subset.

    Hm, now I see that this same point is also amplified in a comment right below.

    So maybe I am missing your point?

    To make sure we are looking at the same part of the entry: You can copy and paste the relevant code from the entry to the Forum here. (Just make sure you check the radio button “Markdown+Itex” below the edit pane here.)

    • CommentRowNumber4.
    • CommentAuthorVitalyR
    • CommentTimeFeb 10th 2021
    Sorry I should refer to Proposition 2.2 in the last comment. I remove "contained in" in Proposition 2.2, see the diff:
    https://ncatlab.org/nlab/revision/diff/interactions+of+images+and+pre-images+with+unions+and+intersections/17

    Now everything there should be OK.

    I love nLab! Thanks for your great work!
    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeNov 30th 2021
    It seems to me that the equality in Proposition 4.1 requires the function, f, to be injective. An arbitrary function, f, gives us that the RHS is a subset of the LHS.
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2021
    • (edited Nov 30th 2021)

    Just to highlight that the question is about the “projection formula” here which was apparently added by Todd Trimble in rev 10 (leaving the direction under question now as “to be continued”).

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2021

    For what it’s worth, the statement in question (here) is also stated as Exercise A.4(k) in Lee 2000.

    diff, v19, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2022

    I have slightly adjusted formatting and wording of the remark on inverse image being right adjoint to direct image (here)

    diff, v20, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2023

    added (here) also a mentioning of the Beck-Chevalley condition satisied by forming pre-/images

    (essentially copied over from what I just added there)

    diff, v22, current

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