In classical algebraic topology, we have the following important relationship between cohomology and homotopy:
$H^{n}(X,G)\approx[X,K(G,n)]$
where $K(G,n)$ is an Eilenberg-Maclane space, characterized by the property that only its nth homotopy group is nontrivial and it is isomorphic to $G$.
How does this generalize to other cohomology theories, in particular sheaf cohomology? I have found "Abstract Homotopy and Generalized Sheaf Cohomology" by Brown, which states that
$H^{n}(X,F)\approx[\mathbf{Z},F_{(n)}]_{\text{Ho }\mathcal{C}(X)}$
where $\mathbf{Z}$ and $F_{(n)}$ are complexes of sheaves, with $\mathbf{Z}$ concentrated in dimension zero, consisting of the constant sheaf with stalk equal to the group of integers., while $F_{(n)}$ is the sheaf $F$ concentrated in dimension $-n$.
I looked at the article "cohomology" from the nLab, which states what is possibly the most general definition of cohomology, but I do not yet grasp how $\mathbf{Z}$ and $F_{(n)}$ in the category of complexes of abelian sheaves are analogous respectively to $X$ and $K(G,n)$ in the category of topological spaces.