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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2016

    I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 6th 2020

    Added redirects.

    diff, v27, current

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeJan 13th 2021

    Added the change-of-base Quillen equivalence between over-categories for suitable weak equivalences between the bases.

    diff, v30, current

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeJun 17th 2021
    • (edited Jun 17th 2021)

    The two categories C and D had got mixed up at one place, so η d:dR(L(d))\eta_d:d\to R(L(d)) was there instead of η c:cR(L(c))\eta_c:c\to R(L(c)). I think it is right now. (I checked in the reference given and they seem to have the mistake as well. Confusing.)

    In fact this part (proposition 2.3) of the entry seems to be in bit of a mess, as RR and LL or CC and DD have somehow ended up being swapped from earlier on in the entry. I am confused!!!

    diff, v32, current

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeAug 20th 2021

    The given sufficient condition for C/SC/TC/S \leftrightarrows C/T to be a Quillen equivalence is actually a necessary condition too.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2021
    • (edited Aug 20th 2021)

    Then let’s make this part of the statement, not hide it in the proof.

    I have reworked both the statement and the proof a fair bit (here), adding more explanation throughout (the key use of 2-out-of-3 wasn’t even mentioned before).

    Also I cross-linked with base change Quillen adjunction. The whole proposition would rather be found there than here, and so I am copying it over now.

    diff, v34, current

  1. Fixed typo in Proposition 2.3


    diff, v36, current

  2. The first condition mentioned in the proof of proposition 2.3 (#InCaseTheRightAdjointCreatesWeakEquivalences) contained an erroneous R.


    diff, v37, current

    • CommentRowNumber9.
    • CommentAuthorGuest
    • CommentTimeFeb 9th 2023
    Does anyone know the original paper for Quillen equivalence?
    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 9th 2023

    Re #9: The original reference for Quillen equivalences is Quillen’s Homotopical Algebra, see there for bibliographic data.