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    • 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.