I was wondering wether anyone had any examples as to why it more useful to consider a sheaf theory approach to TDA problems.
I am familiar with some of the benefits of using cellular cosheaves to recover the global persistence module of a point cloud from local persistence modules as is done in https://arxiv.org/pdf/2001.01623.pdf
Particularly interesting would be examples where it is clear that the standard persistence theory (i.e considering the persistence module after applying the simplicial homology functor) fails to work but sheaf theory perspective (i.e. applying cosheaf homology or sheaf cohomology) works.
I have however not found any results for uses of actual sheaves. Is it ever useful to consider persistence sheaf cohomology?
