Magnitude meets persistence: homology theories for filtered simplicial sets

The Euler characteristic is an invariant of a topological space that in precise sense captures its canonical notion size, akin to the cardinality set. closely related homology space, as it can be expressed alternating sum Betti numbers, whenever well-defined. Thus, one says categorifies characteristic. In his work on generalisation cardinality-like invariants, Leinster introduced magnitude metric real number gives effective points space. Recently, and Shulman theory for spaces, called homology, which When studying often only interested up rescaling distance by non-negative number. function describes how changes scales distance, completely encoded homology. finite data analysis using persistent approximates through nested sequence simplicial complexes so recover information about this sequence. Here we relate two different ways computing filtered sets.