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nLab > Latest Changes: bar construction

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2010
    • (edited Nov 24th 2010)

    the standard bar complex of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.

    I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).

    (That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2015

    If I see correctly, all occurences of “Δ\Delta” at bar construction refer to the augmented simplex category, which elsewhere is denoted “Δ +\Delta_+” or “Δ a\Delta_a”. I vote for changing the notation to one of these notations. While it is true that this will add a little bit of notational overhead to the entry, it serves to avoid confusion for all readers who happen to jump into the entry at some point and miss some paragraph at the beginning.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 20th 2015
    • (edited Oct 20th 2015)

    I changed all the Δ\Delta’s in bar construction to Δ a\Delta_a’s (and introduced the notation at the exact point where augmented simplex category of mentioned).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2015

    Thanks, Todd!!

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeFeb 4th 2020
    Hello;

    Seems to be a slight infelicity in wording if I am not mistaken. The author refers to a "comonad map" from P to |-|; I see that this is a natural transformation but I don't think (unless I am reading it incorrectly) this commutes with counit and comultiplication, as one sometimes uses comonad map or morphism of comonads to mean. Perhaps just natural transfomation?
    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 4th 2020

    I agree; there’s a mistake there. I’ll fix it a bit later.

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeFeb 9th 2020
    My comment concerns this email
    http://www.math.ucr.edu/home/baez/trimble/bar.html
    rather than this page per se, but I assume it is ok to post here, as the page seems to be based on the content of the email.

    I do not understand the geometric intuition for why we can think of the left adjoint to the decalage comonad as a "cone" monad. I don't see how to prove that it agrees with the mapping cone when we pass to the geometric realization.

    I posted a more detailed version of this question to math.stackexchange.com as well. Any help would be appreciated, or a reference where this is worked out in some detail.
    https://math.stackexchange.com/questions/3539674/cone-monad-on-simplicial-sets

    Thanks very much for your time!
    - Patrick N.
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 9th 2020

    Hi Patrick,

    welcome to the nnLab!

    Just to say that to make your comments come out with intended formatting, choose “Mardown+Itex” below the edit box. Then things here work mostly as on the nLab itself.

    In particular, for quotation marks just type

      "

    as usual, and for hyperlinks do

      [link text](url)
    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 9th 2020

    Hi Patrick,

    Those notes were written a long time ago, and I didn’t know then that John would be posting them publicly (not that I really mind). You are right that the end result in TopTop (or in your favorite convenient category of spaces) should be called the mapping cylinder of the canonical quotient X→π 0(X)X \to \pi_0(X), not the mapping cone. It amounts to a coproduct of cones, one for each connected component x∈π 0(X)x \in \pi_0(X). Does this help, or do you need something more?

  10. Add ref to sec 3.6 of Weibel.

    Ramkumar Ramachandra

    diff, v30, current

    • CommentRowNumber11.
    • CommentAuthorPaoloPerrone
    • CommentTimeOct 7th 2020

    Added a reference

    diff, v31, current

  12. Add ref to Homology IV.5.

    Ramkumar Ramachandra

    diff, v32, current

    • CommentRowNumber13.
    • CommentAuthorTim_Porter
    • CommentTimeNov 6th 2021

    Added some related concepts

    diff, v33, current

    • CommentRowNumber14.
    • CommentAuthorjim stasheff
    • CommentTimeAug 4th 2023
    It would be good to add a link where the bar construction is related to the W bar construction
    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 9th 2024

    Added:

    The original reference for bar constructions in the generality of monads is

    diff, v34, current

    • CommentRowNumber16.
    • CommentAuthorJohn Baez
    • CommentTimeJun 11th 2024

    I tried to sketch the basic idea of the bar resolution in the first section.

    diff, v38, current

    • CommentRowNumber17.
    • CommentAuthorJohn Baez
    • CommentTimeJun 12th 2024

    Added heuristic explanation at the very start.

    diff, v41, current

    • CommentRowNumber18.
    • CommentAuthorJohn Baez
    • CommentTimeJun 12th 2024

    Included formula for face maps in EGE G. Please check indices and numbering conventions.

    diff, v45, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2024

    added various pointers to simplicial classifying space, where this is discussed.

    diff, v46, current

1 to 19 of 19

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