Recall the case of cohomology of a topological space with values in a sheaf of abelian Groups. Let $X$ be a topological space and $\mathcal{F}$ be a sheaf on $X$. We want to introduce sheaf cohomology of $\mathcal{X}$ with values in $\mathcal{F}$. We introduce global section functor $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ and then, as $\Gamma(X,-)$ is left exact functor, it produces right derived functors $R^i\Gamma(X,-)$ and $R^i\Gamma(X,\mathcal{F})$ is declared as $i$th cohomology of $X$ with valued in sheaf $\mathcal{F}$ and we denoted this by $H^i(X,\mathcal{F})$. (Correct me if I have misunderstood something till here).

Recall the case of cohomology of a topological space with values in a sheaf of abelian Groups. Let $X$ be a topological space and $\mathcal{F}$ be a sheaf on $X$. We want to introduce sheaf cohomology of $X$ with values in $\mathcal{F}$. We introduce global section functor $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ and then, as $\Gamma(X,-)$ is left exact functor, it produces right derived functors $R^i\Gamma(X,-)$ and $R^i\Gamma(X,\mathcal{F})$ is declared as $i$th cohomology of $X$ with valued in sheaf $\mathcal{F}$ and we denoted this by $H^i(X,\mathcal{F})$. (Correct me if I have misunderstood something till here).

Recall the case of cohomology of a topological space with values in a sheaf of abelian Groups. Let $X$ be a topological space and $\mathcal{F}$ be a sheaf on $X$. We want to introduce sheaf cohomology of $\mathcal{X}$ with values in $\mathcal{F}$. We introduce global section functor $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ and then, as $\Gamma(X,-)$ is left exact functor, it produces right derived functors $R^i\Gamma(X,-)$ and $R^i\Gamma(X,\mathcal{F})$ is declared as $i$th cohomology of $X$ with valued in sheaf $\mathcal{F}$ and we denoted this by $H^i(X,\mathcal{F})$. (Correc me if I have misunderstood something till here).

Let $\mathcal{X}\rightarrow Man$ be a differentiable stack. Fixing a Grothendieck topology on the category $Man$ induces a Grothendieck topology on $\mathcal{X}$. Thus, we can talk about sheaf over $\mathcal{X}$ just like a sheaf over a topological space. Sheaf over $\mathcal{X}$ is a functor $\mathcal{F}:\mathcal{X}^{op}\rightarrow (Set)$.

To imitate the notion of definition of sheaf cohomology in topological set up, we need to make sense of notion of Global section functor. $$\Gamma(\mathcal{X},-):(\text{sheaves over } \mathcal{X})\rightarrow (Sets)$$

Given a stack $\mathcal{X}$ they define a sheaf $\Omega^i_{\mathcal{X}}$ for each $i$ as some type of differential forms and then say, this gives a complex $\Omega$ of sheaves of $\mathbb{R}$ vector spaces over $\mathcal{X}$ which they call as deRham complex of $\mathcal{X}$. They say, its Hypercohomology (defined above for a complex $\mathcal{M}$) is called the deRham cohomology of $\mathcal{X}$:

I am trying to relate this notion of defining deRham cohomology using sheaf cohomology with the same thing happening i.e., going from sheaf cohomology to deRham cohomology in case of manifold.

Recall the case of cohomology of a topological space with values in a sheaf of abelian Groups. Let $X$ be a topological space and $\mathcal{F}$ be a sheaf on $X$. We want to introduce sheaf cohomology of $\mathcal{X}$ with values in $\mathcal{F}$. We introduce global section functor $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ and then, as $\Gamma(X,-)$ is left exact functor, it produces right derived functors $R^i\Gamma(X,-)$ and $R^i\Gamma(X,\mathcal{F})$ is declared as $i$th cohomology of $X$ with valued in sheaf $\mathcal{F}$ and we denoted this by $H^i(X,\mathcal{F})$. (Correct me if I have misunderstood something till here).

Let $\mathcal{X}\rightarrow Man$ be a differentiable stack. Fixing a Grothendieck topology on the category $Man$ induces a Grothendieck topology on $\mathcal{X}$. Thus, we can talk about sheaf over $\mathcal{X}$ just like a sheaf over a topological space. Sheaf over $\mathcal{X}$ is a functor $\mathcal{F}:\mathcal{X}^{op}\rightarrow (\text{Sets})$.

To imitate the notion of definition of sheaf cohomology in topological set up, we need to make sense of notion of Global section functor. $$\Gamma(\mathcal{X},-):(\text{sheaves over } \mathcal{X})\rightarrow (\text{Sets})$$

Given a stack $\mathcal{X}$ they define a sheaf $\Omega^i_{\mathcal{X}}$ for each $i$ as some type of differential forms and then say, this gives a complex $\Omega$ of sheaves of $\mathbb{R}$ vector spaces over $\mathcal{X}$ which they call as de Rham complex of $\mathcal{X}$. They say, its Hypercohomology (defined above for a complex $\mathcal{M}$) is called the de Rham cohomology of $\mathcal{X}$:

I am trying to relate this notion of defining de Rham cohomology using sheaf cohomology with the same thing happening i.e., going from sheaf cohomology to de Rham cohomology in case of manifold.

I am reading https://arxiv.org/pdf/math/0605694.pdf to understand about (hyper)cohomology groups of stack $\mathcal{X}$ with valued in a complex of abelian sheaves $\mathcal{M}$.

Let $\mathcal{F}$ be a sheaf on a topological space $X$ and we want to introduce sheaf cohomology of $\mathcal{X}$ with values in $\mathcal{F}$. We introduce global section functor $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ and then, as $\Gamma(X,-)$ is left exact functor, it produces right derived functors $R^i\Gamma(X,-)$ and $R^i\Gamma(X,\mathcal{F})$ is declared as $i$th cohomology of $X$ with valued in sheaf $\mathcal{F}$ and we denoted this by $H^i(X,\mathcal{F})$. (Please correct me if I have misunderstood something till here).

Now, we want to generalize the notion of topological space to not just a category (site) but a stack $\mathcal{X}\rightarrow Man$. Fixing a Grothendieck topology on category $Man$ of smooth manifolds induces a Grothendieck topology on $\mathcal{X}$. Thus, we can talk about sheaf over $\mathcal{X}$ just like a sheaf over a topological space. Sheaf over $\mathcal{X}$ is a functor $\mathcal{F}:\mathcal{X}^{op}\rightarrow (Set)$.

I am reading Differentiable Stacks and Gerbes to understand about (hyper) cohomology groups of a stack $\mathcal{X}$ with values in a complex $\mathcal{M}$ of abelian sheaves over $\mathcal{X}$.

Recall the case of cohomology of a topological space with values in a sheaf of abelian Groups. Let $X$ be a topological space and $\mathcal{F}$ be a sheaf on $X$. We want to introduce sheaf cohomology of $\mathcal{X}$ with values in $\mathcal{F}$. We introduce global section functor $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ and then, as $\Gamma(X,-)$ is left exact functor, it produces right derived functors $R^i\Gamma(X,-)$ and $R^i\Gamma(X,\mathcal{F})$ is declared as $i$th cohomology of $X$ with valued in sheaf $\mathcal{F}$ and we denoted this by $H^i(X,\mathcal{F})$. (Correc me if I have misunderstood something till here).

Let $\mathcal{X}\rightarrow Man$ be a differentiable stack. Fixing a Grothendieck topology on the category $Man$ induces a Grothendieck topology on $\mathcal{X}$. Thus, we can talk about sheaf over $\mathcal{X}$ just like a sheaf over a topological space. Sheaf over $\mathcal{X}$ is a functor $\mathcal{F}:\mathcal{X}^{op}\rightarrow (Set)$.