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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 28th 2010

    tried to polish a little and slightly expand model category, starting with the Definition-section and ending with the (new and tiny) Properties section. Added some more subsections and so on.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 7th 2012
    • (edited Mar 21st 2012)

    I have added to model category a section Redundancy in the axioms with a statement of the fact that a model structure is determined by its cofibrations and (just) the fibrant objects. Included a pointer to a proof by Joyal.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2012

    I have now spelled out in the entry the (easy) proof of the above assertion.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2016

    I have spelled out the proof that the non-standard definition used in the entry implies that weak equivalences are closed under retracts here. Also added more explicit pointers to Joyal’s text for this statement.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJul 8th 2016

    Thanks! This is good to have recorded.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 2nd 2017

    At model category it said

    to obtain a quasicategory, given a model category MM, the simplicial nerve N Δ(M cf)N_\Delta(M_{cf}) of the subcategory M cfMM_{cf}\subset M of cofibrant-fibrant objects is a quasicategory.

    John Baez pointed out to me by email that this is wrong; we need the homotopy coherent nerve of the simplicial category of fibrant-cofibrant objects (which I presume is what whoever wrote this sentence had in mind). I fixed the page.

    • CommentRowNumber7.
    • CommentAuthorJohn Baez
    • CommentTimeDec 5th 2018
    • (edited Dec 5th 2018)

    The definition of model category in the page model category doesn’t include the condition that acyclic cofibrations have the left lifting property with respect to fibrations and cofibrations have the left lifting property with respect to acyclic fibrations. Is this right? Later in the article it says

    It is clear that:


    Given a model category structure, any two of the three classes of special morphisms (cofibrations, fibrations, weak equivalences) determine the third:

    • given WW and CC, we have F=RLP(WC)F = RLP(W \cap C);

    • given WW and FF, we have C=LLP(WF)C = LLP(W \cap F);

    • given CC and FF, we find WW as the class of morphisms which factor into a morphism in CWC \cap W followed by a morphism in FWF \cap W.

    The terms RLP and LLP are not explained.

    On a separate note, I’m confused about how the third item above lets us find WW given CC and FF, because it’s circular. Is this item attempting to say that WW is the unique subcategory such that any morphism in that subcategory factors as a morphism in CWC \cap W followed by a morphism in FWF \cap W? Or maybe that given CC and FF there’s a unique model structure for which the weak equivalences have this property?

    • CommentRowNumber8.
    • CommentAuthorRuneHaugseng
    • CommentTimeDec 5th 2018

    The condition that acyclic cofibrations have the left lifting property for fibrations, etc., is packaged up in the definition of “weak factorization system” here (condition 2 in Definition 2.1).

    A non-circular way of interpreting the description of WW is to read CWC \cap W here as being defined as acyclic cofibrations, meaning morphisms with the left lifting property for fibrations, which is determined once you specify FF.

    • CommentRowNumber9.
    • CommentAuthorJohn Baez
    • CommentTimeDec 5th 2018

    Okay, I get it. I think this should be explained better.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeDec 5th 2018

    Feel free to explain it better! (-:

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2018

    where “RLP” appears first, I added this remark:

    (Here RLP(S)RLP(S) denotes the class of morphisms with the right lifting property against SS and LLP(S)LLP(S) denotes the class of morphisms with the left lifting property against SS.)

    diff, v78, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2018
    • (edited Dec 5th 2018)

    Until somebody has the energy to rewrite this entry into something more inviting, I have added the following line at the beginning of the Definition-section:

    The following is a somewhat terse account. For a more detailed exposition see at Introduction to Homotopy Theory the section Abstract homotopy theory.

    diff, v78, current

    • CommentRowNumber13.
    • CommentAuthorMarc
    • CommentTimeDec 6th 2018

    Just changed the explanation as suggested in #8

    diff, v79, current

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 2nd 2020

    Added all books that treat model categories at length.

    diff, v83, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 2nd 2020

    Added a coverage table.

    Feedback and additions welcome!

    diff, v84, current

    • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 2nd 2020

    I would also appreciate additional suggestions for the coverage table. I think it could be a useful resource.

    Perhaps it could also be placed (as a template?) in other articles?

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2020

    added pointer to premodel category under “Related concepts”

    diff, v85, current

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 8th 2020

    Corrected some history.

    Added other weaker notions.

    diff, v86, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2020

    made the new pointer to “Smith” point to Jeff Smith.

    diff, v87, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2020

    Have added ISBNs to the books by Hovey and by Hirschhorn

    diff, v93, current

  1. Added in new reference book “A handbook of model categories” by S. Balchin


    diff, v105, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2022

    Thanks! I had not seen this yet:

    Am giving the author a category: people-page…

    diff, v107, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2023

    Renamed the Properties-subsection “Opposite model structure” to “Basic closure properties of class of model categories” (here) and added the examples of product model structure and transfer of model structure along adjoint equivalences of underlying categories.

    diff, v116, current