Ephemeral Modules and Scott Sheaves in Persistence

Since Justin Curry’s pioneering dissertation, sheaf theory has gained more attention in the field of topological data analysis. In this thesis, we want to offer a new direction for these methods by applying domain theory. In domain theory, a poset has a number of topologies associated to it; the Alexandrov and Scott topologies being the most significant for our investigations. The sheaves on these two topologies and their interplay are our main object of study.

In topological data analysis, features of the data that have a short “lifespan” are often seen as a result of noise, and so they are considered less significant. The ephemeral features are the most extreme examples of this phenomenon having lifespan zero. Chazal and his co-authors introduced the notion of an ephemeral module, but they only considered the poset R.

We generalize the notion of ephemerality to continuous posets. We consider the quotient category of the category of persistence modules by the Serre subcategory of ephemeral modules and show that it is equivalent to the category of Scott sheaves. We prove that these categories are further equivalent to the full subcategories of lower and upper semi-continuous modules. This quotient category has a hereditary torsion associated to it. We examine the corresponding torsion functor, observing in particular that there exists an exact sequence and isomorphisms mimicking the celebrated Serre-Grothendieck correspondence between sheaf cohomology and local cohomology. Finally, we study the metric properties of persistence modules and Scott sheaves via the aforementioned equivalence. We also investigate how ephemeral modules fit into this setting.