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nLab > Latest Changes: Kan fibrant replacement

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 4th 2019

    Added more properties and applications.

    diff, v16, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 8th 2020

    Added redirects “Ex^∞” and “Ex-infinity”.

    diff, v19, current

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeApr 9th 2020
    • (edited Apr 9th 2020)
    The article proposes

    Ex^∞(f) is a simplicial weak equivalences if and only if f is a simplicial homotopy equivalence.

    Isn't that incorrect? It's clear that the statement

    Ex^∞(f) is a weak equivalence in the classical model structure if and only if f is a weak equivalence in the classical model structure

    is true. But, AFAIK, the class of homotopy equivalences expressed using the cylinder X+X -> X×Δ[1] does not coincide with the class of weak equivalences. (if these do coincide, it's surprisingly hard to find this fact explicitly stated...)

    Was the statement meant the other way around? i.e.

    Ex^∞(f) is a simplicial homotopy equivalences if and only if f is a simplicial weak equivalence.

    is also true, since Ex^∞(f) is a morphism between fibrant objects, so it is a homotopy equivalence iff it is a weak equivalence.


    I'm assuming "simplicial homotopy equivalence" means you have simplicial homotopies fg=>1 and 1=>gf.

    I'm assuming the term "simplicial weak equivalence" means a weak equivalence in the classical model structure, but AFAICT that term is barely used on the ncatlab and is never defined.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2020

    Please feel invited to edit the entry to clarify/fix.

    Recently I made “simplicial weak equivalence” redirect to classical model structure on simplicial sets, but it would deserve it’s own entry.

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeApr 9th 2020
    I'll take that to mean I've said nothing factually inaccurate, in which case I'm happy to make the change.
    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeApr 9th 2020

    Fixed an error in the phrasing of the property that Ex^\infty reflects weak equivalences.

    diff, v20, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 11th 2020

    It was meant to be the other way around, of course.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2023

    added pointer to the analog version of ExEx for the (localization of) quasi-categories incarnated as marked simplicial sets:

    diff, v23, current

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 26th 2023

    Re #8: It seems to me these authors only develop an analogue of Ex, not Ex^∞.

    Their main theorem seems to say that if a marked quasicategory satisfies the calculus of left fractions, then the analogue of the functor Ex sends it to a quasicategory.

    As far as I can see, there is nothing about fibrantly replacing arbitrary marked simplicial sets, so there is no analogue of Ex^∞.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2023

    Absolutely, it’s an “analogue version of ExEx”, as per #8.

    diff, v24, current

1 to 10 of 10

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