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nLab > Latest Changes: homotopy limit

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 22nd 2019

    Formatting fix

    diff, v93, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 27th 2022
    • (edited Aug 27th 2022)

    Added:

    Terminology

    In older literature, such as the original work of Bousfield–Kan \cite{BousfieldKan}, the homotopy limit of a diagram X:I→sSetX\colon I\to sSet is defined as the weighted limit

    hom(I/−,X),hom(I/-, X),

    where the weight I/−I/- sends i↦N(I/i)i\mapsto N(I/i), the nerve of the comma category of ii.

    Such a functor must be derived in order to get the correct (homotopy invariant) notion, which in this case amounts to replacing XX with an objectwise weakly equivalent diagram valued in Kan complexes.

    At some point (when?) a shift in terminology happened, and in the modern parlance homotopy limits are commonly assumed to be derived.

    diff, v100, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2022

    I have added to this addition (here) a link to Bousfield-Kan map and also to the entry’s own section “Resolved (co)ends”.

    diff, v101, current

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMar 13th 2023

    Corrected obsolete links to updated ones in

    I, localizers, (pdf);

    II, model categories, (pdf);

    III, derivators, (pdf):

    IV, summary on derivators (pdf)

    diff, v103, current

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 5th 2023

    Added:

    diff, v108, current

    • CommentRowNumber6.
    • CommentAuthorvarkor
    • CommentTimeNov 16th 2023

    Added a reference to A general formulation of homotopy limits.

    diff, v109, current

1 to 6 of 6

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