In the last few years there have been efforts made to generalise the theory of peristence homology and cohomology to deal with sheaves for example Russold - Persistent sheaf cohomology. This as i understand is based on work Sheaves, Cosheaves and Applications done by Curry where he develops cellular sheaves and cosheaves.
One application he lists is the time dependent sensor evasion problem (section 10 of the linked paper). In other words we have a space $X$ (usually manifold with boundary) and a subspace $U_t$ — the detection region, which varies with parameter $t\in [0,1]$. We want to know whether there exists a path joining $X\setminus U_0\times \{0\}$ to $X\setminus U_1\times \{1\}$ in $X\times [0,1]$. Curry proposes to analyse the problem using cosheaves or sheaves (after linearising the sheaf of sections via Reeb graphs) and looking at persistent zero dimensional cohomologies. However he concludes that his approach in both cases is not successful in extracting whether there exists an evasion path. That is there exist evasion paths even when 0-dimensial homologies persist.
I was wondering whether there have been any successful applications of the theory of persistence homology with sheaf coefficient either to this problem or another one and separately in which scenarios it is better to use sheaf cohomology vs cosheaf homology in practice (with examples if such exist please).
