Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 Xπ 0(X)X \to \pi_0(X), not the mapping cone. It amounts to a coproduct of cones, one for each connected component xπ 0(X)x \in \pi_0(X). Does this help, or do you need something more?

  1. Add ref to sec 3.6 of Weibel.

    Ramkumar Ramachandra

    diff, v30, current

    • CommentRowNumber11.
    • CommentAuthorPaoloPerrone
    • CommentTimeOct 7th 2020

    Added a reference

    diff, v31, current

  2. 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