One Diamond to Rule Them All: Old and new topics about zigzag, levelsets and extended persistence

Extended and zigzag persistence were introduced more than ten years ago, as generalizations of ordinary persistence. While overcoming certain limitations of ordinary persistence, they both enjoy nice computational properties, which make them an intermediate between ordinary and multi-parameter persistence, with already existing efficient software implementations. Nevertheless, their algebraic theory is more intricate, and in the case of extended persistence, was formulated only very recently. In this context, this paper presents a richly illustrated self-contained introduction to the foundational aspects of the topic, with an eye towards recent applications in which they are involved, such as computational sheaf theory and multi-parameter persistence.