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    • CommentRowNumber1.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMar 27th 2011
    • (edited Mar 27th 2011)

    do we already have this in nLab? it seems that the long exact sequence in cohomology

    H n(X,Y;A)H n(X,A)H n(Y,A)H n+1(X,Y;A) \cdots \to H^n(X,Y;A)\to H^n(X,A)\to H^n(Y,A) \to H^{n+1}(X,Y;A)\to \cdots

    for an inclusion YXY\hookrightarrow X should have the following very simple and natural interpretation: for a morphism f:YXf:Y\to X in a (oo,1)-topos H\mathbf{H} and a coefficient object AA together with a fixed morphism φ:YA\varphi:Y\to A, consider the induced morphism f *:H(X,A)H(Y,A)f^*:\mathbf{H}(X,A)\to \mathbf{H}(Y,A) and take its (homotopy) fiber over the point *φH(Y,A)*\stackrel{\varphi}{\to}\mathbf{H}(Y,A). In particular, when the coefficient object AA is pointed, we can consider the case where φ:YA\varphi:Y\to A is the distinguished point of H(Y,A)\mathbf{H}(Y,A). In this case the homotopy fiber one is considering should be denoted H(X,Y;A)\mathbf{H}(X,Y;A) and is the hom-space for the cohomology of the pair (X,Y)(X,Y) with coefficients in AA (here one should actually make an explicit reference to the morphism f:YXf:Y\to X in the notation, unless it is “clear” as in the case of the inclusion of the classical cohomology of a pair). then, for a deloopable coefficients object AA, the long exact sequence in cohomology should immediately follow from the fiber sequence

    H(X,Y;A) H(X,A) * H(Y,A) \array{ \mathbf{H}(X,Y;A) &\to& \mathbf{H}(X,A) \\ \downarrow && \downarrow \\ * &\to& \mathbf{H}(Y,A) }
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 28th 2011

    Hi Domenico,

    yes, that looks right.

    I think I once started something on “relative cohomology” somewhere, but never followed up on it. So this is not on the nnLab currently, as far as I know. But it should be!

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeMar 28th 2011
    • (edited Mar 28th 2011)

    Domenico, the relative cohomology for a map ii denoting the embedding is in many Urs’s manuscripts including in the nactwist http://www.math.uni-hamburg.de/home/schreiber/nactwist.pdf. Very little is there for comparison though.

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