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nLab > Latest Changes: strict ind-object

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  1. Exposé in SGA4 was not mentioned

    Anonymous

    diff, v4, current

  2. Exposé in SGA4 was not mentioned

    Anonymous

    diff, v4, current

  3. Exposé in SGA4 was not mentioned

    Anonymous

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeFeb 21st 2020

    I do not understand this entry. If an object is isomorphic to a colimit it is itself a colimit of the same type of the diagram, namely the original cone postcomposed with the isomorphism. So, what is the imaginary difference between strict ind-objects and “essentially strict”. I understand that there are some concrete presentations of ind objects as diagrams themselves, then there is a difference between those which are strict on the nose and those which are isomorphic, but this is not according to the definition in the entry which considers ind objects as colimits somewhere (even, there is “vertex” mentioned!). Besides, it is not in the spirit of nn-Lab to make definitions depending on very specific presentations. I have never heard of essentially strict objects despite working much on the strict ind-pro-objects with a PhD student.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 21st 2020

    When I just looked, I saw “essentially monomorphic”, but not “essentially strict”.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2020
    • (edited Feb 22nd 2020)

    Looks like this is about the paragraph added in revision 3 (by Zoran! :-)

    I read this as that the clause “which is isomorphic to” is just being misleading and could be removed: The paragraph is simply saying that “strict ind-objects are also called essentially monomorphic objects”.

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 22nd 2020

    I deleted the redundant paragraph about isomorphisms.

    diff, v6, current

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeMar 3rd 2020
    • (edited Mar 3rd 2020)

    Thank you Urs, 6! Your solution to the wording is good.

    I do not remember writing this, but it must be me, as I did indeed research the subject in 2017, when the paragraph was recorded, maybe I came across some confusing reference at that point (and forgot about it quickly, as I claim above to never hear of it!). But with present level of piece of the mind at least I clearly see the added paragraph misleading.

    Thanks again.

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