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nLab > Latest Changes: lax (∞,1)-colimit

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  1.  
    • CommentRowNumber1.
    • CommentAuthorHurkyl
    • CommentTimeFeb 25th 2021

    I added the description of lax (co)limits of Cat-valued functors via (co)ends and ordinary (co)limits. I should probably flesh this out more.

    I’ve adopted the convention on twisted arrows at twisted arrow category, which is opposite of that in GNN.

    In the case of ordinary 2-category, when the diagram category is a 1-category, is the expression of lax (co)limits via ordinary weighted (co)limits really as simple as taking the weights C •/C_{\bullet/} or C /•C_{/\bullet}? I can’t find a reference that spells that out clearly; if there really is such a simple description it should be put on the lax (co)limit page.

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeFeb 25th 2021

    Corrected an error, so I changed it to something true until I sort out how to agree with the existing convention on nLab.

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeMar 5th 2021

    Forgot to remove the reference to the twisted arrow category after removing it from the equivalent formulas

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeMar 5th 2021

    Added in the oplax variants. Also corrected an error where I had defined the lax colimit with the oplax formula.

    diff, v8, current

  5. Added the example of the lax limit of an arrow.

    Anonymous

    diff, v9, current

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeMar 25th 2021

    Decided it was also worth mentioning the lax colimits over an arrow.

    diff, v11, current

    • CommentRowNumber7.
    • CommentAuthorHurkyl
    • CommentTimeMar 27th 2021

    Clarified the functorality of the slices being used.

    diff, v13, current

    • CommentRowNumber8.
    • CommentAuthorHurkyl
    • CommentTimeOct 20th 2021

    Fixed the variance on the correspondence to the contravariant version. To go along with it, added the (op)lax colimits of a constant functor as an example.

    diff, v14, current

    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeOct 20th 2021

    Ack, I had the variance right the first time, so I’ve reverted that. But I kept the added example and clarification.

    diff, v14, current

    • CommentRowNumber10.
    • CommentAuthorHurkyl
    • CommentTimeJul 6th 2022

    Added a computation of how (op)lax (co)limits are expressed via the Grothendieck construction, and connected the calculation to the intuition that, e.g., functors between cocartesian fibrations (that don’t necessarily preserve cocartesian morphisms) can be thought of as corresponding to lax natural transformations.

    diff, v16, current

1 to 10 of 10

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