Before revision:  Under this framework, certain  well-known summaries, such as persistent Betti numbers, and persistence landscape, are special cases of the PC . Moreover, we prove a rigorous bound for a general families of PCs.
After revision:  Under this framework, certain well-known summaries, such as Betti number curves, the Euler Characteristic Curve, and Persistence Landscapes fall under the PC framework or are easily derived from it. Second, we provide a theoretical foundation for the stability analysis of PCs.