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Cohomology groups of this cochain complex is precisely the singular cohomology groups $H^k(X,A)$. … Thus, sheaf cohomology groups $H^k(X,\underline{A}_X)$ are isomorphic to singular cohomology groups $H^k(X,A)$. …

If $X$ is paracompact and locally contractible, then singular cohomology and Cech cohomology of $X$ coincide, with coefficients in any abelian group. …

groups gives cohomology of the Lie group? … By cohomology of the Lie group, I mean the cohomology of the underlying manifold. I do not think there is any notion of cohomology of Lie group. Even Google search does not give anything. …

For example, "Poisson cohomology and its derivatives are not calculated explicitly even for dual spaces to semisimple Lie algebra". … Question : What is the current status regarding computation of Poisson cohomology groups of semisimple Lie algebras? …

I know one or two things about torsion groups, examples of cohomology groups that has non zero torsion part, some definitions $K$-theory. … Can some one help me to understand why torsion in cohomology should remind about computing $K$-theory? Any references are welcome. …

We have the notion of de Rham cohomology of a Lie groupoid by considering de Rham cohomology of simplicial manifold associated to it, denoted by $\mathcal{G}_\bullet$. … cohomology of the Lie group $G$ and the de Rham cohomology of the manifold $M$? …

The $k$-th cohomology of this double complex is defined to be the $k$-th deRham cohomology of the Lie groupoid $[\mathcal{G}_1\rightrightarrows \mathcal{G}_0]$. … Are there other models that compute cohomology of Lie groupoids? Are there other cohomology theories in the category of Lie groupoids? …

Proposition $13$ and Remark $16$ in page $10$ of Cohomology or Stacks says that, there is a natural isomorphism $$H^i_G(X)\rightarrow H^i(X\times G\rightrightarrows X)$$ where, $H^i_G(X)$ is the $i^{ … \text{th}}$ equivariant cohomology of $X$ with respect to action of $G$ and $H^i(X\times G\rightrightarrows X)$ is the $i^{\text{th}}$ deRham cohomology of the transformation groupoid $(X\times G\rightrightarrows …

Same is the situation with cohomology. Suppose I have a topological space $X$, I can talk about its singular cochain complex and the corresponding singular cohomology. … It is a standard result that, with correct coefficients, singular cohomology is same as deRham cohomology (deRham’s theorem). …

More generally, are cohomology functors sheaves in general (in any reasonably non trivial Grothendieck topology)? … I am also interested in cohomology functors that arise in Algebriac geometry/topology. Is there a way of sheafification in this setup? …