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nLab > Latest Changes: totalization

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2011

    stub for totalization

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeAug 9th 2011
    • (edited Aug 9th 2011)

    Is this related to (one of the two) totalization(s) of a double complex in homological algebra and related totalizations in homological algebra ?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2011

    Good point, the terminology is a bit inconclusive in some places of the literature, I think. I have added the following paragraph to totalization:

    Formally the dual to totalization is geometric realization: where totalization is the end over a powering with Δ\Delta, realization is the coend over the tensoring.

    But various other operations carry names similar to “totalization”. For instance a total chain complex is related under Dold-Kan correspondence to the diagonal of a bisimplicial set – see Eilenberg-Zilber theorem. As discussed at bisimplicial set, this is weakly homotopy equivalent to the operation that is often called TotTot and called the total simplicial set of a bisimplicial set.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeAug 10th 2011

    Whereas the diagonal of a bisimplicial set is actually its geometric realization, considered as a simplicial object in sSet!

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeAug 11th 2011
    • (edited Aug 11th 2011)
    • Paul Bressler, Alexander Gorokhovsky, Ryszard Nest, Boris Tsygan, Deformation quantization of gerbes, Advances in Mathematics 214 (2007) 230–266, pdf

    reviews the constructon of the totalization of a cosimplicial DGLA in Section 3.4 (defined as certain colimit) and “prove that isomorphism classes of descent data of a cosimplicial DGLA are in one-to-one correspondence with isomorphism classes of Maurer–Cartan elements of its totalization”.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2018
    • (edited Sep 24th 2018)

    added statement and pointer to proof of the fact that the totalization, in the traditional sense, is a model for the homotopy limit over the cosimplicial objects, if the latter is Reedy fibrant.

    diff, v6, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2018

    also added brief mentioning of Bousfield-Kan spectral sequence and Eilenberg-Moore spectral sequence.

    (phew, that entry had been sitting there all the years lacking most of the homotopy theoretic information…)

    diff, v6, current

  8. Previous wording somewhat dismissive.

    Rosona Eldred

    diff, v8, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2023

    Thanks for the heads-up. I have added date and author-link (here).

    I can’t recall why I wrote “some kind of notes” back in 2011, when this was the first and only reference for the entry, but I am sure it was not meant dismissively. Probably a combination of thinking this “might be lecture notes or might be a preprint, hard to tell” mixed with bad English skills.

    diff, v9, current

  10. Given ya’ll uploaded your personal copy, removing the link to my google drive.

    Rosona Eldred

    diff, v11, current

1 to 10 of 10

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