Realisations of posets and tameness

Persistent homology is commonly encoded by functors with values in the category of vector spaces and indexed by a poset. These functors are called tame or persistence modules and capture the life-span of homological features in a dataset. Every poset can be used to index a persistence module, however some of them are particularly well suited. We introduce a new construction called realisation, which transforms posets into posets. Intuitively it associates a continuous structure to a locally discrete poset by filling in empty spaces. Realisations share several properties with upper semilattices. They behave similarly with respect to certain notions of dimension for posets that we introduce. Moreover, as indexing posets of persistence modules, they both have good discretisations and allow effective computation of homological invariants via Koszul complexes.

This talk is based on a joint work with Wojciech Chachólski and Alvin Jin.