We propose sparse versions of filtered simplicial complexes used to compte persistent homology of point clouds and of networks. In particular we extend a slight variation of the Sparse \v{ C}ech Complex of Cavanna, Jahanseir and Sheehy from point clouds in Cartesian space to point clouds in arbitrary metric spaces. Along the way we formulate interleaving in terms of strict $2$-categories, and we introduce the concept of Dowker dissimilarities that can be considered as a common generalization of metric spaces and networks.
Categories: math.AT