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 https://arxiv.org/pdf/2204.13446.pdf. This as i understand is based on work done by Curry where he develops cellular sheaves and cosheaves https://arxiv.org/pdf/1303.3255.pdf.

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 wether 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 succesful in extracting whether there exists an evasion path. That is there exist evasion paths even when 0-dimensial homologies persist.

I was wondering wether there have been any succesful application of the the theory of persitence homology with sheaf coefficient either to this problem or another one and more separately in which scenarios is it better to use shaf cohomology vs cosheaf homology in practice (with examples if such exist please).