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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2009
    • (edited Nov 23rd 2009)

    I expanded proper model category a bit.

    In particular I added statement and (simple) proof that in a left proper model category pushouts along cofibrations out of cofibrants are homotopy pushouts. This is at Proper model category -- properties

    On page 9 here Clark Barwick supposedly proves the stronger statement that pushouts along all cofibrations in a left proper model category are homotopy pushouts, but for the time being I am failing to follow his proof.


    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2009
    • (edited Nov 23rd 2009)
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <p>I wrote:</p> <blockquote> but for the time being I am failing to follow his proof. </blockquote> <p>Oh, I get it. I was being stupid. Will add the statement and proof now.</p> </div>
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2009
    • (edited Nov 23rd 2009)

    okay, here now -- in full beauty -- the proof that pushouts along cofibrations in left proper model categories are homotopy pushouts.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2010

    Added more examples and counter-examples to proper model category.

    Stated Charles Rezk's theorem about passing to proper Quillen equivalent models for simplicial algebras over simplicial theories. Also stated Thomas Nikolaus' theorem about Quillen equivalent models of fibrant objects (which are in particular right proper).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2013

    Made the statement that “all objects (co)fibrant” implies (left)right properness more explicit in Properties and added a citation.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 7th 2016

    I added to proper model category the result discussed at this MO question that to prove right properness we are free to assume the base object of the pullback to be fibrant.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 20th 2017
    • (edited Feb 20th 2017)

    Which of the standard dg-algebra categories are proper, such as dg-(co)algebras or dg-Lie algebras?

    [edit: I see here on MO discussion that the projective model structure on unbounded dgc-algebras is proper.]

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJan 14th 2019

    Mentioned in the definition section that an apparently-weaker condition suffices, with a link to the proposition below.

    diff, v36, current

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJan 14th 2019

    BTW, there is a bug in this page: the link “General” from the table of contents does not work. I think the problem is that there is also a heading hidden in the “Context” menu called “general”, and so that’s the one that gets linked to. Is there anything that we can do about this? E.g. could the software detect duplicate header names and disambiguate them somehow, say with numbers?

  1. Thanks for raising this. I will try to remember to look into it when I get the chance. It will not be for at least a few days.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJan 30th 2019

    Added publication data for Every homotopy theory of simplicial algebras admits a proper model, and cited it for the characterization of right properness in terms of Quillen equivalence of slice categories.

    diff, v40, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 30th 2019
    • (edited Jan 30th 2019)

    changed this slightly, from


    diff, v41, current

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 17th 2020

    References for counterexamples.

    diff, v42, current

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 7th 2020


    diff, v43, current

    • CommentRowNumber15.
    • CommentAuthorGuest
    • CommentTimeMay 24th 2020
    Error in the examples section. Claims that every simplicially enriched category is cofibrant in the model structure. This is not true. That in fact is the main technical advantage in working with quasi-categories instead of simplicially enriched categories (e.g. see the Appendix of HTT, and the section in the first chapter on homotopy-coherent diagrams). (Julian)
    • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 24th 2020

    Added by Urs Schreiber in Revision 8 on March 25, 2010.

    If the standard Dwyer-Kan-Bergner model structure is used, then few simplicial categories are cofibrant.

    I am not sure if one can enlarge the class of cofibrations to all (local) monomorphisms (say), which would make all objects cofibrant, but it seems difficult: acyclic cofibrations must be closed under cobase changes, and the latter are rather complicated in simplicial categories.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2020
    • (edited May 25th 2020)

    Thanks for the alert. Checking, I don’t find the paragraph which this is about. I guess you guys fixed it? But I also don’t see it in that revision 8 of “proper model category”.

    (BTW, you can link to specific points in an entry by adding an anchor.)

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 25th 2020
    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2020
    • (edited May 25th 2020)


    So I have replaced the line with

    the model structure for quasi-categories

    Checking whether the error was induced from an error at model structure on sSet-categories… But that speaks of right-properness (here)

    diff, v44, current

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 30th 2022


    The Rezk criterion for properness

    The following criterion shows that the notion of left or right properness only depends on the underlying relative category of a model category, i.e., does not depend on fibrations or cofibrations. This is clear once we observe that the notion of a Quillen equivalence in the statement below can be replaced by the notion of a Dwyer–Kan equivalence of underlying relative categories.

    \begin{theorem} (Rezk \cite{Rezk02}, Proposition 2.7 (arXiv), Proposition 2.5 (journal).) A model category MM is left proper if and only if for every weak equivalence f:XYf\colon X\to Y the induced Quillen adjunction

    X/MY/MX/M\leftrightarrows Y/M

    is a Quillen equivalence. A model category MM is right proper if and only if for every weak equivalence f:XYf\colon X\to Y the induced Quillen adjunction

    M/XM/YM/X\leftrightarrows M/Y

    is a Quillen equivalence. \end{theorem}

    diff, v51, current

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