Persistent homology

Homological features are tracked across a nested family of spaces or data scales.

Different ways to understand it

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Cohomology
Dual cochain constructions yield invariants and additional operations such as cup products.
Higher homotopy groups
Based maps from spheres modulo homotopy measure higher-dimensional homotopy structure.
Homology groups
Cycles modulo boundaries measure holes through a chain complex.
Chain complexes
Successive boundary maps compose to zero in a graded sequence of groups or modules.
Covering spaces
A covering locally maps disjoint sheets homeomorphically onto an open neighborhood.