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
    • CommentTimeDec 29th 2011
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2011
    • (edited Dec 29th 2011)

    I have added a section over an ∞-group with the statement that for BB an \infty-group object, the \infty-pullback of any f:XBf : X \to B along any g:YBg : Y \to B is the homotopy fiber of fgf - g (the difference in BB).

    This follows easily if BB is such that

    B * Δ B 0 B×B B \array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{\mathrlap{0}} \\ B \times B &\stackrel{-}{\to}& B }

    is an \infty-pullback (the bottom morphism is “take the difference” here written additively as for abelian \infty-groups).

    This ought to be true generally, but currenty in the entry I give the argument only for the special case that BB has a presentation in the image of the Dold-Kan correspondence.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2011

    This ought to be true generally

    I have now added the claim and proof for this statement in any 1-localic \infty-topos.

    • CommentRowNumber4.
    • CommentAuthorjim_stasheff
    • CommentTimeDec 31st 2011
    That's NOT the Mayer-Vietoris sequence of my youth which was for homology NOT homotopical. is that innovation due to
    Dyer et al? The one I know and love is a very useful alternative axiom for the LES in Eilenberg-Steenrod.
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 31st 2011
    • (edited Dec 31st 2011)

    The (co)homological MV sequence is a special case of the homotopical one, in view of the fact that the homotopy groups of a simplicial group are the (co)homology groups of the corresponding complex, by Dold-Kan.

    I have added to the entry now an Examples-section with a subsection (Co)Homology of a cover where I indicate how the historically first example that I guess you have in mind is reproduced as a special case.

    is that innovation due to Dyer et al?

    I don’t know for sure if this is the first reference that gives the bigger picture, but it is the first that I have found so far. I had originally stated the more general version at fiber sequence without further ado, because I considered it obvious. Then a fews days back I wanted to spell out the details, as I have done now, and googled around a bit to see if anyone had stated this before. I found that Dyer et al did. But I wouldn’t be surprised to hear that this has been clear to others before.