Categorification of Persistent Homology

@article{Bubenik2014CategorificationOP,
  title={Categorification of Persistent Homology},
  author={Peter Bubenik and J. Scott},
  journal={Discrete & Computational Geometry},
  year={2014},
  volume={51},
  pages={600-627}
}
  • Peter Bubenik, J. Scott
  • Published 2014
  • Mathematics, Computer Science
  • Discrete & Computational Geometry
  • We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are $\mathbf {(\mathbb {R},\leq)}$-indexed diagrams in some target category. A set of such diagrams has an interleaving distance, which we show generalizes the previously studied bottleneck distance. To illustrate the utility of this approach, we generalize previous stability results for persistence, extended persistence, and kernel, image, and cokernel persistence. We give a… CONTINUE READING
    124 Citations
    Generalized persistence diagrams
    • Amit Patel
    • Mathematics, Computer Science
    • J. Appl. Comput. Topol.
    • 2018
    • 36
    • PDF
    Persistence modules: Algebra and algorithms
    • 10
    • PDF
    Persistent homology and microlocal sheaf theory
    • 39
    • PDF
    Metrics for Generalized Persistence Modules
    • 88
    • PDF
    A comparison framework for interleaved persistence modules
    • 8
    • PDF
    Death and extended persistence in computational algebraic topology
    • Highly Influenced
    • PDF
    Persistence Modules on Commutative Ladders of Finite Type
    • 31
    • PDF
    Induced Matchings of Barcodes and the Algebraic Stability of Persistence
    • 50
    • Highly Influenced
    • PDF
    Stable Signatures for Dynamic Metric Spaces via Zigzag Persistent Homology
    • 9

    References

    SHOWING 1-10 OF 49 REFERENCES
    Extending Persistence Using Poincaré and Lefschetz Duality
    • 108
    • PDF
    Metrics for Generalized Persistence Modules
    • 88
    • PDF
    Multidimensional Interleavings and Applications to Topological Inference
    • 22
    • PDF
    Proximity of persistence modules and their diagrams
    • 333
    • PDF
    Persistent homology for kernels, images, and cokernels
    • 45
    • PDF
    The Structure and Stability of Persistence Modules
    • 276
    • PDF
    Semi-abelian categories
    • 420
    Stability of persistence diagrams
    • 819
    • PDF
    Zigzag Persistence
    • 192
    • PDF
    Barcodes: The persistent topology of data
    • 783
    • PDF