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    • CommentRowNumber1.
    • CommentAuthorZhen Lin
    • CommentTimeJun 29th 2014

    I added the remark that the canonical model structure on Cat is the model structure obtained by transferring the projective model structure on bisimplicial sets.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 29th 2014

    That statement sounded very confusing until I looked at the page and realized you were transferring it along Rezk’s classifying diagram functor!

    Surely you’ll get the same thing if you transfer the injective/Reedy model structure?

  1. There is a unique model structure on Cat with the weak equivalences the equivalences of categories (at least assuming AC). So if these other variants transfer then they give the same model structure.

    • CommentRowNumber4.
    • CommentAuthorZhen Lin
    • CommentTimeJun 29th 2014

    Huh. That’s surprising – for instance, Beke has shown that there are different choices of cofibrations in sSetsSet for the usual weak homotopy equivalences in sSetsSet. How does one prove uniqueness for CatCat?

  2. Another nice exercise along these lines is to show (again assuming AC) that the category of sets has precisely nine Quillen model structures, no more, no less.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 30th 2014

    Chris, that’s just great; thank you! I’ve now mentioned this at canonical model structure on Cat (just something quick for now, with links). I hope a fuller treatment will appear at some point within the nLab (and really this result deserves to be shouted from the rooftops; I had not heard it either).

    • CommentRowNumber8.
    • CommentAuthorZhen Lin
    • CommentTimeJul 1st 2014

    Very nice result! I suppose this is the ultimate justification for the name ‘canonical’.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 2nd 2014

    I went ahead and did a write-up of the uniqueness result, pretty much exactly Chris’s proof with only a few minor changes.

    • CommentRowNumber10.
    • CommentAuthorCharles Rezk
    • CommentTimeApr 9th 2015
    • (edited Apr 9th 2015)

    I added a reference to the earliest reference for the canonical model structure (although only for 1-groupoids) that I am aware of.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 10th 2015

    @Charles where did you add this? Neither canonical model structure on groupoids or canonical model structure on Cat show recent edits. Is this reference prior to

    D.W. Anderson, Fibrations and Geometric Realizations, Bull. Am. Math Soc. 84, 765-786, (1978), 765-786.

    as shown at canonical model structure on groupoids?

    • CommentRowNumber12.
    • CommentAuthorCharles Rezk
    • CommentTimeApr 11th 2015

    Oh, apparently I added it to canonical model structure, which seems not to be cross-linked with the pages you mention.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2015

    Thanks for the alert. I have added the missing cross-links.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2015

    Looking at the entry made me move all the references to the bottom of the entry and edit a bit to make it more standard. (new References section).

    Also, I notice that the pointer to Anderson has been there are model structure on Grpd all along. Everbody who cares about the quality of the nLab: please take this incident as further incentive to be mindful about adding sufficient cross-links.

    • CommentRowNumber15.
    • CommentAuthorJohn Baez
    • CommentTimeOct 10th 2019

    I added a brief comparison to the Thomason model structure, just to make it easy for readers to find that page.

    diff, v20, current

    • CommentRowNumber16.
    • CommentAuthorRodMcGuire
    • CommentTimeOct 10th 2019

    The link in

    Along similar lines, one can prove (assuming AC) that there are nine – count ’em, nine – Quillen model structures on SetSet.

    is broken.

    That page is available via Wayback

    but its math symbols are broken.

    However Omar Antolín Camarena is now in Mexico and hosts the page there so I swapped in the link

    diff, v21, current

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 21st 2020

    Added a reference to CatLab.

    diff, v23, current

    • CommentRowNumber18.
    • CommentAuthorTim_Porter
    • CommentTimeDec 3rd 2020
    • (edited Dec 3rd 2020)

    Changed ’different than’ to ’different from’ which is easier on the ear!

    diff, v24, current

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 3rd 2020

    I believe that’s a UK/US thing, Tim. They do insist on speaking differently from/than us.

    • CommentRowNumber20.
    • CommentAuthorTim_Porter
    • CommentTimeDec 3rd 2020

    There are occasions even in English where ’different’ does take ’than’, but the one I changed did not seem to be one of them. (I loved the announcement over the intercom on a US flight that ‘the plane will momentarily be landing’! Very confusing for an English speaker. Perfectly clear for an Unitedstatesian speaker. (I do not know the correct translation of the French term ’étatsunian’. ) There should be a good term for the language spoken in the US, and which is closely related to English. :-)

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeNov 1st 2023

    added missing pointer to:

    By the way, does any publication state the cartesian monoidal model property of the canonical model structure on CatCat?

    diff, v26, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeNov 1st 2023
    • (edited Nov 2nd 2023)

    added statement and proof (here) that the canonical model structure is cartesian monoidal.

    In the course of this I have re-arranged the rest of the entry somewhat by moving the propositions out of the Definition-section into a Properties-section.

    diff, v27, current

    • CommentRowNumber23.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 2nd 2023


    diff, v29, current

    • CommentRowNumber24.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 2nd 2023

    @Tim #20

    I’ve seen USAnian applied my a mathematician to the people; it might work for the language too, I guess.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2023

    the original discussion by Joyal & Tierney is already in the generality of internal categories (internal to topoi). So I have changed the wording on the item Everaert et al. to clarify that they generalize this to finitely complete ambient categories.

    diff, v30, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2023

    stated the observation (under “General properties”, here) that all objects are bifibrant so that the model structure is proper.

    diff, v31, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2023

    added pointer to

    and added the statement of cofibrant generation (here)

    diff, v32, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2023

    For completeness, I added some words on the cofibrant generation statement (here).

    diff, v33, current