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do we already have this in nLab? it seems that the long exact sequence in cohomology
for an inclusion should have the following very simple and natural interpretation: for a morphism in a (oo,1)-topos and a coefficient object together with a fixed morphism , consider the induced morphism and take its (homotopy) fiber over the point . In particular, when the coefficient object is pointed, we can consider the case where is the distinguished point of . In this case the homotopy fiber one is considering should be denoted and is the hom-space for the cohomology of the pair with coefficients in (here one should actually make an explicit reference to the morphism 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 , the long exact sequence in cohomology should immediately follow from the fiber sequence
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 Lab currently, as far as I know. But it should be!
Domenico, the relative cohomology for a map 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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