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Let $X$ be the $2$-sphere. Let $\mathcal D$ be the bounded derived category of sheaves over $X$. Let $\underline{\mathbb C}_X$ be the constant sheaf over $X$ with stalks isomorphic to $\mathbb C$.

Then why is $\text{Hom}_{\mathcal D}(\underline{\mathbb C}_X,\underline{\mathbb C}_X[2]) \cong \mathbb C$?

The morphism in the derived category is defined to be the equivalence class of "roofs". By this definition, I should find "roofs" like $\underline{\mathbb C}_X \overset{q}{\leftarrow} F \overset{u}{\rightarrow} \underline{\mathbb C}_X[2]$, where $q$ is a quasi-isomorphism, and then determine the equivalence of them. But I have no idea how I can do this.

Through the comments under this post, I know that $\text{Hom}_{\mathcal D}(\underline{\mathbb C}_X,\underline{\mathbb C}_X[2]) =\text{Ext}^2(\underline{\mathbb C}_X, \underline{\mathbb C}_X)$, calculated in the category of sheaves over $X$. But I still don't know why this is true from the definition.

More generally, if $A,B$ in $\mathcal D$ do not just concentrate in one degree, how can I better understand and calulate $\text{Hom}_{\mathcal D}(A,B)$?

Many thanks!

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    Not a full answer, but in general you have $\operatorname{Hom}_{\mathcal{D}}(\underline{\mathbb{C}}_X,\mathcal{F}[p])=H^p(X,\mathcal{F})$. So in your example, you have $\operatorname{Hom}_{\mathcal{D}}(\underline{\mathbb{C}}_{S^2},\underline{\mathbb{C}}_{S^2}[2])=H^2(S^2,\mathbb{C})=\mathbb{C}$. (Here I supposed that $\mathcal{D}$ is the derived category of sheaves of $\mathbb{C}$-vector spaces). – 2017-02-09
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    @Roland: Thank you very much! As $\text{Hom}_{\mathcal D}(\underline{\mathbb C}_X,\underline{\mathbb C}_X[2]) =\text{Ext}^2(\underline{\mathbb C}_X, \underline{\mathbb C}_X) = H^2(X, \underline{\mathbb C}_X)$, I am wondering how to connect these three things. – 2017-02-10
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    What do you mean by connect ? Do you want to know why these equalities are true ? – 2017-02-10
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    @Roland: Yes, exactly! – 2017-02-10
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    Well, the first equality should be in any book on the derived category. The second is just the fact that $\operatorname{Ext}^p(\underline{\mathbb C}_X,.)$ and $H^p(X,.)$ are the derived functor of $\operatorname{Hom}(\underline{\mathbb C}_X,.)=\Gamma(X,.)$ (these two functors are naturally isomorphic). – 2017-02-10

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