Poincar\'e Duality for Generalized Persistence Diagrams of (co)Filtrations

TL;DR

This paper extends the concept of generalized persistence diagrams to cofiltrations, establishing a Poincaré duality relationship on manifolds and exploring the functoriality and connections to Rota's Galois Connection Theorem.

Contribution

It introduces a dualization of generalized persistence diagrams for cofiltrations and links this duality to Poincaré duality on manifolds, emphasizing functoriality and theoretical connections.

Findings

Established Poincaré duality between persistence diagrams of filtrations and cofiltrations on manifolds.

Demonstrated functoriality of generalized persistence diagrams.

Connected the theory to Rota's Galois Connection Theorem.

Abstract

We dualize previous work on generalized persistence diagrams for filtrations to cofiltrations. When the underlying space is a manifold, we express this duality as a Poincar\'e duality between their generalized persistence diagrams. A heavy emphasis is placed on the recent discovery of functoriality of the generalized persistence diagram and its connection to Rota's Galois Connection Theorem.