Explore a comprehensive framework for topological summarization in this 57-minute lecture by Tamal Dey, presented for the Applied Algebraic Topology Network. Delve into key concepts such as covers, nerves, and maps between covers, before progressing to more advanced topics like pullback covers and hierarchical covering complexes. Learn about the Multiscale Mapper and its persistence diagram, as well as the stability of Hierarchical Function-Ordered Simplicial Structures (HFOSS) against perturbations. Examine various cover examples and stability against function perturbations, with a focus on PL functions and combinatorial Multiscale Mapper. Gain insights into the metric viewpoint and Čech filtrations using du before concluding with a Q&A session. This talk provides a thorough exploration of topological summarization techniques and their applications in algebraic topology.