Topological spaces of persistence modules and their properties

  • Peter BubenikEmail author
  • Tane Vergili


Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules, including many of those that have been previously studied, and describe the relationships between them. In the cases where these classes are sets, interleaving distance induces a topology. We undertake a systematic study the resulting topological spaces and their basic topological properties.


Persistent homology Persistence modules Interleaving distance 

Mathematics Subject Classification

55N99 54D99 54E99 18A25 



The authors would like to that the anonymous referees for their helpful suggestions. In particular, we would like to thank the referee who contributed the proof that the enveloping distance from pointwise-finite dimensional persistence modules to q-tame persistence modules is zero. We also thank Alex Elchesen for proofreading an earlier draft of the paper. The first author would like to acknowledge the support of UFII SEED funds, ARO Research Award W911NF1810307, and the Southeast Center for Mathematics and Biology, an NSF-Simons Research Center for Mathematics of Complex Biological Systems, under National Science Foundation Grant No. DMS-1764406 and Simons Foundation Grant No. 594594.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Supplementary material


  1. Bauer, U., Lesnick, M.: Induced matchings and the algebraic stability of persistence barcodes. J. Comput. Geom. 6(2), 162–191 (2015)MathSciNetzbMATHGoogle Scholar
  2. Bauer, U., Lesnick, M.: Persistence diagrams as diagrams: a categorification of the stability theorem (2016). arXiv preprint arXiv:1610.10085
  3. Bjerkevik, H.B., Botnan, M.B.: Computational complexity of the interleaving distance. In: 34th International Symposium on Computational Geometry, vol. 12 (2017). arXiv preprint arXiv:1712.04281
  4. Blumberg, A.J., Lesnick, M.: Universality of the homotopy interleaving distance (2017). arXiv preprint arXiv:1705.01690
  5. Blumberg, A.J., Gal, I., Mandell, M.A., Pancia, M.: Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces. Found. Comput. Math. 14(4), 745–789 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Botnan, M., Lesnick, M.: Algebraic stability of zigzag persistence modules. Algebra Geom. Topol. 18(6), 3133–3204 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bubenik, P.: Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16, 77–102 (2015)MathSciNetzbMATHGoogle Scholar
  8. Bubenik, P., Dlotko, P.: A persistence landscapes toolbox for topological statistics. J. Symb. Comput. 78, 91–114 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bubenik, P., Scott, J.A.: Categorification of persistent homology. Discrete Comput. Geom. 51(3), 600–627 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Bubenik, P., de Silva, V., Scott, J.: Metrics for generalized persistence modules. Found. Comput. Math. 15(6), 1501–1531 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bubenik, P., de Silva, V., Nanda, V.: Higher interpolation and extension for persistence modules. SIAM J. Appl. Algebra Geom. 1(1), 272–284 (2017a)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Bubenik, P., de Silva, V., Scott, J.: Interleaving and Gromov–Hausdorff distance and interleaving of functors (2017b). arXiv preprint arXiv:1707.06288
  13. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discrete Comput. Geom. 42(1), 71–93 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Chazal, F., Michel, B.: An introduction to topological data analysis: fundamental and practical aspects for data scientists (2017). arXiv preprint arXiv:1710.04019
  16. Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, vol. 09, ACM, New York, NY, USA, pp. 237–246 (2009)Google Scholar
  17. Chazal, F., de Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedic. 173, 193–214 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Chazal, F., Crawley-Boevey, W., de Silva, V.: The observable structure of persistence modules. Homol. Homotopy Appl. 18(2), 247–265 (2016a)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. Springer Briefs in Mathematics. Springer, Cham (2016b)CrossRefzbMATHGoogle Scholar
  20. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Collins, A., Zomorodian, A., Carlsson, G., Guibas, L.J.: A barcode shape descriptor for curve point cloud data. Comput. Gr. 28(6), 881–894 (2004)CrossRefGoogle Scholar
  22. Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. J. Algebra Appl. 14(5): 1550066, 8 (2015)MathSciNetzbMATHGoogle Scholar
  23. Curry, J.: Sheaves, cosheaves and applications. PhD Thesis, University of Pennsylvania (2014)Google Scholar
  24. de Silva, V., Munch, E., Patel, A.: Categorified Reeb graphs. Discrete Comput. Geom. 55(4), 854–906 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. de Silva, V., Munch, E., Stefanou, A.: Theory of interleavings on \([0,\infty )\)-actegories (2017). arXiv preprint arXiv:1706.04095
  26. Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscr. Math. 6, 71–103 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. (N.S.) 45(1), 61–75 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Ghrist, R.: Homological algebra and data. Math. Data 25, 273 (2018)Google Scholar
  29. Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Found. Comput. Math. 15(3), 613–650 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Meehan, K., Meyer, D.: Interleaving distance as a limit (2017a). arXiv preprint arXiv:1710.11489
  31. Meehan, K., Meyer, D.: An isometry theorem for generalized persistence modules (2017b). arXiv preprint arXiv:1710.02858
  32. Mileyko, Y., Mukherjee, S., Harer, J.: Probability measures on the space of persistence diagrams. Inverse Probl. 27(12): 124007, 22 (2011)MathSciNetzbMATHGoogle Scholar
  33. Munch, E., Wang, B.: Convergence between categorical representations of Reeb space and mapper (2015). arXiv preprint arXiv:1512.04108
  34. Munkres, J.R.: Topology: A First Course. Prentice-Hall Inc., Englewood Cliffs (1975)zbMATHGoogle Scholar
  35. Patel, A.: Generalized persistence diagrams. J. Appl. Comput. Topol. 1(3), 397–419 (2018)CrossRefzbMATHGoogle Scholar
  36. Puuska, V.: Erosion distance for generalized persistence modules (2017). arXiv preprint arXiv:1710.01577
  37. Turner, K., Mileyko, Y., Mukherjee, S., Harer, J.: Fréchet means for distributions of persistence diagrams. Discrete Comput. Geom. 52(1), 44–70 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. Wasserman, L.: Topological data analysis (2016). arXiv preprint arXiv:1609.08227 [stat.ME]
  39. Webb, C.: Decomposition of graded modules. Proc. Am. Math. Soc. 94(4), 565–571 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
  2. 2.Department of MathematicsEge UniversityIzmirTurkey

Personalised recommendations