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nLab > Latest Changes: Grothendieck pretopology

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 21st 2021

    Replaced the requirement of existence of all pullbacks with the requirement that pullbacks of covering families exist, to match SGA 4.

    Added a reference to SGA 4.

    diff, v21, current

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 23rd 2021

    I have no idea why the stronger definition was there for so long. It’s just not true in applications, most particularly the pretopology consisting of surjective submersions on the category of manifolds.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 21st 2022
    • (edited May 10th 2022)

    Need to fix this false statement:

    Notice that a basis for the topology of X is not a Grothendieck pretopology on Op(X) (since it is in general not closed under pullback, which here is restriction) but is a coverage on Op(X).

    The definition of a pretopology does not require morphisms that belong to some covering family to be closed under pullbacks. Rather, it requires that base changes of morphisms in covering families always exist.

    In particular, base changes of inclusions whose domains belong to the base always exist, even though the resulting inclusion has as its domain an open subset that need not belong to the base.

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 21st 2022

    As discussed in this answer

    https://mathoverflow.net/questions/416691/when-is-a-basis-of-a-topological-space-a-grothendieck-pretopology

    there are at least 3 different sites one can produce from a base of a topological space, and only one of them (arguably the least natural one) gives a pretopology.

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 10th 2022

    Corrected a mistake and added another reference.

    diff, v23, current

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