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 $\mathcal{F}:\mathcal{X}^{op}\rightarrow (Set)$.

We consider the category of sheaves over $\mathcal{X}$ just like category of sheaves over $X$.

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

Let $\mathcal{F}$ is a sheaf on $\mathcal{X}$. We do not have (or atleast I do not see) an object in $\mathcal{X}$ that behaves like $X$ where I can just evaluate $\mathcal{F}$ and get a set.

What one can do is, see that there is a bijection $\text{Hom}_{\text{sheaves(X)}}(\underline{0},\mathcal{F})\rightarrow \mathcal{F}(X)$ from here in Vakil's notes.

Thus, $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ with $\mathcal{F}\mapsto \mathcal{F}(X)$ can be seen as $$\Gamma(X,-):(\text{abelian sheaves on } X)\rightarrow (\text{abelian groups})$$ with $\mathcal{F}\mapsto \text{Hom}_{\text{sheaves(X)}}(\underline{0},\mathcal{F})$.

Second way of seeing global sections functor can be extended to stakcs, with out much changes.

$$\Gamma(\mathcal{X},-):(\text{abelian sheaves on } \mathcal{X})\rightarrow (\text{abelian groups})$$ with $\mathcal{F}\mapsto \text{Hom}_{\text{sheaves}(\mathcal{X})}(\underline{0},\mathcal{F})$.

I am guessing this is what they mean in page $18$ https://arxiv.org/pdf/math/0605694.pdf when they say global section functor.

Just for the same reason as global section functor on $X$ is left exact , the global section functor on $\mathcal{X}$ is left exact (I did not check, is there anything non trivial happening here that does not happen in case of just a topological space? I do not think so).

As this functor $\Gamma(\mathcal{X},-)$ is left exact, we can talk about right derived functors $R^i\Gamma(\mathcal{X},-)$ which are also denoted by $$H^i(\mathcal{X},-):(\text{abelian sheaves on } \mathcal{X})\rightarrow (\text{abelian groups})$$

Then the paper says something which I do not quite understand.

Passing to derived category of complexes of abelian sheaves over $\mathcal{X}$, we get the total derived functor $$R\Gamma(\mathcal{X},-):D^+(\mathcal{X})\rightarrow D^+(\text{abelian groups}).$$ For a complex $\mathcal{M}\in D^+(\mathcal{X})$ of abelian sheaves on $\mathcal{X}$, the homology groups of the complex $R\Gamma(\mathcal{X},\mathcal{M})$ are denoted by $\mathbb{H}^i(\mathcal{X},\mathcal{M})=h^i(R\Gamma(\mathcal{X},\mathcal{M}))$ are called the hypercohomology groups of $\mathcal{X}$ with values in $\mathcal{M}$.

I understand almost nothing that is written in above block. I have not seen something similar when studying sheaf cohomology on a scheme (I did not read so much).Can someone point me to similar definition in case of sheaf cohomology on a scheme. What is the necessity to go to so called complex of abelian sheaves? What am I missing here?

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 $\mathcal{X}$:

$$H^i_{DR}(\mathcal{X})=\mathbb{H}^i(\mathcal{X},\Omega_{\mathcal{X}})$$ This is done in page $19$.

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.

Any suggestions regarding this are welcome.