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nLab > Latest Changes: geometric realization of categories

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  1. Added remark on geometric realizations of pairs of adjoint functors

    Roman T

    diff, v17, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 1st 2021

    Added a bunch of redirects.

    diff, v18, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 18th 2021

    In the formulation of Quillen Theorem A, should it not say “initial (∞,1)-functor” instead of “cofinal (∞,1)-functor”? The page initial functor seems to use the same comma category…

    diff, v19, current

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeDec 19th 2021
    • (edited Dec 19th 2021)

    Initial and cofinal mean the same thing. Unless you’re using the other convention where cofinal means final.

    That said, in my opinion using “initial” and “final” is the preferable convention, for no deeper reason that you don’t use the word “cofinal” and force the reader to figure out which of the two opposite conventions you’re using.

    I went to check Higher Topos Theory (which uses the convention that “cofinal means final” – that is, cofinal functors relate to colimits) to make sure the theorem is stated correctly. This is theorem 4.1.3.1. (d↓F)(d \downarrow F) is used in the version for final functors, so the (F↓d)(F \downarrow d) at the nLab page should be the version for initial functors.

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 19th 2021

    Initial and cofinal mean the same thing.

    What book or published article uses “cofinal” to mean “initial”?

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeDec 19th 2021
    • (edited Dec 19th 2021)

    Cisinski’s “Higher Categories and Homotopical Algebra”, Definition 4.4.13 defines a morphism of simplicial sets X→YX \to Y to be cofinal iff the opposite morphism is final.

    (and to make it clear it’s not using “final = initial” that I see mentioned in a nLab article, which I’ve not actually seen before, Cisinski defines a “final object” to be a final functor Δ 0→X\Delta^0 \to X and an “initial object” to be a cofinal functor Δ 0→X\Delta^0 \to X)

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 19th 2021

    Re #6: Thanks, I did not know about this. This is pretty awful! I think the “cofinal” terminology should be retired immediately.

  8. added a subsection on a variation of Quillen’s Theorem B for Grothendieck fibration, and added appropriate references.

    Kensuke Arakawa

    diff, v23, current

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 11th 2024

    Added details of published version:

    diff, v26, current

    • CommentRowNumber10.
    • CommentAuthorncfavier
    • CommentTimeOct 4th 2024

    Could someone clarify definition 3.1? What is the poset of simplices of a simplicial set? This only makes sense for a simplicial complex AFAIK, but the nerve of CC might not be a simplicial complex.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 5th 2024

    Thanks for the alert. This ought to be the poset of non-degenerate simplices, I suppose.

    Have added that word. But this Def and the following Prop need improving, at least a reference.

    (Am on my phone right now, may add something later.)

    diff, v27, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 5th 2024
    • (edited Oct 5th 2024)

    Rather, I suppose one has to take the poset of non-degenerate simplices of the subdivision (by the proposition now recorded here at Barratt nerve).

    I have added a quick fix in the entry here, accordingly. But I don’t really have the leisure for this at the moment, so if anyone feels like editing further, please do.

    diff, v28, current

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