A Sheaf-Theoretic Construction of Shape Space

Explore a sheaf-theoretic approach to constructing shape space in this 53-minute lecture from the Applied Algebraic Topology Network. Delve into the concept of a homotopy sheaf on the poset category of constructible sets, mapping each set to the Persistent Homology Transform (PHT). Discover how recent findings, building on Schapira's work, demonstrate the PHT's injectivity as an effective shape summary. Learn about the process of "gluing" PHTs to construct larger shapes and examine a generalized nerve lemma for polyhedral PHTs. Investigate metrics on shape space and a general stability theorem for homotopic shapes. Conclude by analyzing the Smale-Niyogi-Weinberger sampling result to understand how to approximate a manifold's PHT using polyhedra with arbitrary precision.