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1. • CommentRowNumber1.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMar 27th 2011
   • (edited Mar 27th 2011)
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexdo 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]] $$ \array{ \mathbf{H}(X,Y;A) &\to& \mathbf{H}(X,A) \\ \downarrow && \downarrow \\ * &\to& \mathbf{H}(Y,A) } $$

   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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 28th 2011
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   Author: Urs
   Format: MarkdownItexHi 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!

   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!

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeMar 28th 2011
   • (edited Mar 28th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexDomenico, 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.

   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.