Sometimes a topologial space (X, $T_{open}$) has the property that its closed sets form a topology $T_{closed}$ too (Alexandrov space). In addition one has a sheaf $\mathcal{S}_{open }$ with respect to $T_{open}$ and a cosheaf $\mathcal{S}_{closed}$ on $T_{closed}$ with a certain functorial morphism $F: \mathcal{S}_{open} \rightarrow \mathcal{S}_{closed}.$
My question: Are there papers which study the homology and cohomology of such linked $(sheaf, cosheaf)$-pairs? What's a suitable categorical setting?