Ulrich Bauer - Persistence Diagrams as Diagrams

Explore the categorical perspective of persistence diagrams in this lecture from the Applied Algebraic Topology Network. Delve into functors indexed over the reals and taking values in the category of matchings, examining how this approach yields a categorical structure on barcodes. Learn about the reformulation of the induced matching theorem and its implications for proving algebraic stability of persistence barcodes. Discover an explicit construction of barcodes for pointwise finite-dimensional persistence modules that doesn't require decomposition into indecomposable interval summands. Investigate the functoriality of this construction on monomorphisms and epimorphisms of persistence modules. Follow the journey through topics such as interval decompositions, the category of matchings, bottleneck distance as an interleaving distance, and the categorified induced matching theorem. Gain insights into the structure of persistence sub-quotient modules, algebraic stability, and a general criterion for trivial colkernels. Conclude with an exploration of rank formulas for barcodes and the construction of barcodes from scratch.