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do we already have this in nLab? it seems that the long exact sequence in cohomology
$\cdots \to H^n(X,Y;A)\to H^n(X,A)\to H^n(Y,A) \to H^{n+1}(X,Y;A)\to \cdots$for an inclusion $Y\hookrightarrow X$ should have the following very simple and natural interpretation: for a morphism $f:Y\to X$ in a (oo,1)-topos $\mathbf{H}$ and a coefficient object $A$ together with a fixed morphism $\varphi:Y\to A$, consider the induced morphism $f^*:\mathbf{H}(X,A)\to \mathbf{H}(Y,A)$ and take its (homotopy) fiber over the point $*\stackrel{\varphi}{\to}\mathbf{H}(Y,A)$. In particular, when the coefficient object $A$ is pointed, we can consider the case where $\varphi:Y\to A$ is the distinguished point of $\mathbf{H}(Y,A)$. In this case the homotopy fiber one is considering should be denoted $\mathbf{H}(X,Y;A)$ and is the hom-space for the cohomology of the pair $(X,Y)$ with coefficients in $A$ (here one should actually make an explicit reference to the morphism $f:Y\to X$ 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 $A$, the long exact sequence in cohomology should immediately follow from the fiber sequence
$\array{ \mathbf{H}(X,Y;A) &\to& \mathbf{H}(X,A) \\ \downarrow && \downarrow \\ * &\to& \mathbf{H}(Y,A) }$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 $n$Lab currently, as far as I know. But it should be!
Domenico, the relative cohomology for a map $i$ 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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