Stability of Universal Constructions for Persistent Homology
thesisposted on 01.08.2019, 00:00 by Janis Lazovskis
In this thesis I describe poset stratifications of the product of the Ran space and the nonnegative real numbers, as a universal space for the Cech construction of simplicial complexes. This leads to a cosheaf valued in diagrams of simplicial complexes for which every restriction to a finite collection recovers the persistent homology of the collection. For the stratification, I describe a partial order on isomorphism classes of abstract simplicial complexes, which allows spaces stratified by them to have entrance paths uniquely interpreted as simplicial maps. Decomposing entrance paths gives a sheaf structure, which has higher information that the cosheaf does not capture.