One-dimensional persistent homology is arguably the most important and heavily used tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas. In this talk, we introduce a certain 2-dimensional persistence module structure associated with the persistent cohomology ring, where one parameter is the cup-length and the other is the filtration parameter. We show that this new persistence structure, called the persistent cup module, is stable.
In addition, we consider a generalized notion of persistent invariants, which extends the standard rank invariant, Puuska's rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariants also enables us to lift the LS-category of topological spaces to a novel stable persistent invariant, called the persistent LS-category invariant.