Generalized Persistence and Graded Structures

The correspondence theorem of Carlsson and Zomorodian, which states that one can view persistence modules as modules over a polynomial ring of one variable, opened the graded perspective in topological data analysis. In this thesis, we want to propose a new generic theoretical framework for understanding generalized persistence modules from this perspective by considering monoid actions on preordered sets. Secondly, in the case the indexing set is a poset, we introduce a new tameness condition for a generalized persistence module by defining the notion of S-determinacy, where S is a subposet containing all the ‘births’ and the ‘deaths’.

We first focus on the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the category of additive functors from this category to the category of Abelian groups is isomorphic to the category of modules graded over the set with a monoid action, and to the category of unital modules over a certain smash product.

In the case S is finite, our notion of S-determinacy leads to a new characterization for a generalized persistence module being finitely presented. Moreover, we show that after adding ‘infinitary points’ to Zn, ‘S-determined’ is equivalent to ‘finitely determined’ as defined by Miller.