Let $X$ be a topological space and denote by $Ab(X)$ the category of abelian sheaves on $X$. My question is on the category of simplicial abelian sheaves $[\Delta^{op},Ab(X)]$. A natural way to define a cohomology for these sheaves is to take $$[\Delta^{op},Ab(X)]\overset{[\Delta^{op},R\Gamma(X,-)]}{\longrightarrow}[\Delta^{op},D(Ab)].$$ Is this the "right" definition for the cohomology of simplicial sheaves? For instance, does this functor respect weak equivalences?
I think another natural definition would be to say that the category $[\Delta^{op},Ab(X)]$ is abelian and to consider the the right derived functor of $s\Gamma(X,-):[\Delta^{op},Ab(X)]\to [\Delta^{op},Ab]$ which would yield a right derived functor $$Rs\Gamma(X,-):[\Delta^{op},Ab(X)]\to D\left([\Delta^{op},Ab]\right).$$ How do these two functors compare? Could we have $[\Delta^{op},D(Ab)]\cong D\left([\Delta^{op},Ab]\right)$?