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Concept# Persistence module

Summary

A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological features of an object across a range of scale parameters. A persistence module often consists of a collection of homology groups (or vector spaces if using field coefficients) corresponding to a filtration of topological spaces, and a collection of linear maps induced by the inclusions of the filtration. The concept of a persistence module was first introduced in 2005 as an application of graded modules over polynomial rings, thus importing well-developed algebraic ideas from classical commutative algebra theory to the setting of persistent homology. Since then, persistence modules have been one of the primary algebraic structures studied in the field of applied topology.
Let be a partially ordered set (poset) and let be a field. The poset is sometimes called the indexing set. Then a persistence module is a functor from the poset of to the category of vector spaces over and linear maps. A persistence module indexed by a discrete poset such as the integers can be represented intuitively as a diagram of spaces: To emphasize the indexing set being used, a persistence module indexed by is sometimes called a -persistence module, or simply a -module.
One can alternatively use a set-theoretic definition of a persistence module that is equivalent to the categorical viewpoint: A persistence module is a pair where is a collection of -vector spaces and is a collection of linear maps where for each , such that for any (i.e., all the maps commute).
In the case of a -module where is a single partially ordered set (e.g., , etc.), we say that is a single- or 1-parameter persistence module. However, if is instead a product of totally ordered sets, i.e., for some totally ordered sets , then by endowing with the product partial order given by only if for all , we can define a multiparameter persistence module indexed by .

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