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nLab > Latest Changes: relation between quasi-categories and simplicial categories

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1 to 17 of 17

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2010

    one more remark at relation between quasi-categories and simplicial categories

    (to be expanded...)

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeNov 24th 2020
    Hi,

    I am aware that this entry is quite old and not finished. However, I was wondering if it is a typo or a theorem (and in the latter case, I would be interested in a reference) that in the relation via the homotopy coherent nerve, neither the simplicial category C is assumed to be fibrant nor the unit is derived. I would be curious about any sort of insight here.

    Best,
    Viktoriya Ozornova
    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 24th 2020

    This claim was added by Urs in Revision 7 on February 11, 2010.

    I am pretty sure that the homotopy coherent nerve functor must be derived, since it doesn’t preserve weak equivalence of arbitrary simplicial categories.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    Thanks for the alert. Could you fix it?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    added the qualifiers “derived” and “fibrant”, which were probably missing.

    But this entry still needs to cite actual facts anyways, where it currently says

    (…details/links…)

    diff, v21, current

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 25th 2020
    • (edited Nov 25th 2020)

    Added more details and references. Changed the notation to the one used in Bergner’s book.

    diff, v22, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    Thanks!

    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTimeNov 26th 2020
    Thank you for the clarification!

    Best,
    Viktoriya
    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeNov 5th 2021

    R is fibrant replacement for the Dwyer-Kan-Bergner model structure, not the Joyal one.

    diff, v25, current

    • CommentRowNumber10.
    • CommentAuthorHurkyl
    • CommentTimeNov 5th 2021

    Added in the calculation of ℭ\mathfrak{C} for some nice simplicial sets.

    diff, v26, current

    • CommentRowNumber11.
    • CommentAuthorHurkyl
    • CommentTimeNov 5th 2021
    • (edited Nov 5th 2021)

    I wanted to add the reason I chose the ordering convention I did on the hom-sets is by analogy with lax operations that have the direction [g]⋅[f]→[gf][g] \cdot [f] \to [gf].

    Also, this has analogy with the Duskin nerve, and it is the ordering used in Lurie’s Kerodon, 2.4.3.1 (simplicial path categories), which later uses this when discussing (∞,2)(\infty,2)-categories.

    • CommentRowNumber12.
    • CommentAuthorHurkyl
    • CommentTimeNov 5th 2021

    Somehow, I had completely missed the homotopy coherent nerve page had much of what I wrote (albeit with the opposite ordering convention on homspaces). So I’ve removed the additions, and will probably add in the additional examples over there.

    diff, v26, current

    • CommentRowNumber13.
    • CommentAuthorTim_Porter
    • CommentTimeJan 2nd 2023

    Lightened the wording slightly.

    diff, v28, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2023

    touched wording and formatting, added publication data to references.

    Finally hyperlinked the previously unnamed operation ℭ\mathfrak{C} of rigidification of quasi-categories (entry to be created now).

    diff, v30, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2023

    In particular, it used to say in the references section that the Quillen equivalence “is described in” Bergner (2007), while Lurie (2009) was only found at the end of the list. I have changed that.

    diff, v30, current

    • CommentRowNumber16.
    • CommentAuthorTim_Porter
    • CommentTimeMay 31st 2023

    Fixed a defunct link.

    diff, v34, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023

    so let’s add the publication data proper:

    diff, v35, current

1 to 17 of 17

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