In this talk, we introduce a persistent (co)homology theory for Cayley digraph grading. We give the algebraic structures of Cayleypersistence object. Specifically, we consider the module structure of persistent (co)homology and show the decomposition of a finitely generated Cayleypersistence module. Moreover, we introduce the persistence-cup product on the Cayley-persistence module and study the twisted structure with respect to the persistence-cup product. As an application on manifolds, we show that the persistent (co)homology is closely related to the persistent map of fundamental classes.