In this talk I will present four case studies of sheaves and cosheaves in topological data analysis. The first two are examples of (co)sheaves in the small:
(1) level set persistence---and its efficacious computation via discrete Morse theory---and,
(2) decorated merge trees and Reeb graphs---enriched topological invariants that have enhanced classification power over traditional TDA methods. The second set of examples are focused on (co)sheaves in the large:
(3) understanding the space of merge trees as a stratified map to the space of barcodes and
(4) the development of a new "sheaf of sheaves" that organizes the persistent homology transform over different shapes.