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nLab > Latest Changes: split essentially surjective functor

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  1. Page created, but author did not leave any comments.

    Anonymous

    v1, current

  2. adding section on split essentially surjective functors in homotopy type theory

    Anonymous

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 3rd 2022

    I’m having trouble parsing even the first definition. Why is an object isomorphic to a (disjoint?) union of objects in the image?

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeSep 3rd 2022

    I think somebody took the definition in the HoTT book and tried to translate it into set theory.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2022
    • (edited Sep 3rd 2022)

    To parse this, add the missing parenthesis: What must be meat is not

    p y∈⨄ x∈CF(x)≅yp_y \in \biguplus_{x \in C} F(x) \cong y

    but

    p y∈⨄ x∈C(F(x)≅y)p_y \in \biguplus_{x \in C} \big( F(x) \cong y \big)

    so that under 0-truncation this gives the usual (“merely”) existence statement:

    ∃ x∈C(F(x)≅y)\exists_{x \in C} \big( F(x) \cong y \big).

    I hope somebody finds the energy to polish-up this entry.

  6. added missing round brackets to definitions

    Anonymous

    diff, v3, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2022

    I’d suggest to also either define or else replace the notation \biguplus: The common usage of this notation is for disjoint union in set-theoretic contexts, where the crucial distinction which this entry is (or ought to be) making isn’t visible.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeSep 3rd 2022

    Fixed some notation, clarified some phrasing. Also removed several false statements:

    • The implications between equivalences and ff+eso were written backwards.
    • The statement about gaunt categories was wrong (I just fixed it there too).
    • In HoTT, even for univalent categories an eso functor may not be split eso, even if you assume AC.

    Anonymous, please back off on these questionable rapid-fire edits. I don’t have the time to check everything you’re writing, and a lot of it is misleading, confusing, or wrong. I very much appreciate your enthusiasm for bringing HoTT into the mainstream of the nLab, but I think a better way forwards right now would be to propose particular changes on the nForum first, and only go ahead with an edit if there is general agreement. If the current situation continues, I may have to suggest some IP blocking.

    diff, v4, current

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