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.

]]>Dyer et al? The one I know and love is a very useful alternative axiom for the LES in Eilenberg-Steenrod. ]]>

This ought to be true generally

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

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

This follows easily if $B$ is such that

$\array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{\mathrlap{0}} \\ B \times B &\stackrel{-}{\to}& B }$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 $B$ has a presentation in the image of the Dold-Kan correspondence.

]]>created Mayer-Vietoris sequence

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