Skip to main content

Persistent homology and microlocal sheaf theory


We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory. For that purpose we study the derived category of sheaves on a real finite-dimensional vector space \(\mathbb {V}\). By using the operation of convolution, we introduce a pseudo-distance on this category and prove in particular a stability result for direct images. Then we assume that \(\mathbb {V}\) is endowed with a closed convex proper cone \(\gamma \) with non empty interior and study \(\gamma \)-sheaves, that is, constructible sheaves with microsupport contained in the antipodal to the polar cone (equivalently, constructible sheaves for the \(\gamma \)-topology). We prove that such sheaves may be approximated (for the pseudo-distance) by “piecewise linear” \(\gamma \)-sheaves. Finally we show that these last sheaves are constant on stratifications by \(\gamma \)-locally closed sets, an analogue of barcodes in higher dimension.

This is a preview of subscription content, access via your institution.


  1. 1.

    \(\mu {\mathrm{supp}}(F)\) was denoted by \({\text {SS}}(F)\) in Kashiwara and Schapira (1990).

  2. 2.

    See Sect. 2.4 for related constructions.

  3. 3.

    In practice the cone \(\gamma \) will be polyhedral.

  4. 4.

    As already mentioned, these results were clarified during discussions of the second named author with Nicolas Berkouk.

  5. 5.

    This statement is due to Curry (2013, Th.4.2.10).


  1. Bauer, U., Lesnick, M.: Persistent diagram as diagrams: a categorification of the stability theorem (2016). arXiv:1610.10085

  2. Bubenik, P., Scott, J.: Categorification of persistent homology. Discrete Comput. Geom. 51, 600–627 (2014). arXiv:1205.3669

    MathSciNet  Article  Google Scholar 

  3. Bubenik, P., de Silva, V., Scott, J.: Metrics for generalized persistence modules. Found. Comput. Math. 15, 1501-1531 (2015). arXiv:1312.3829

    MathSciNet  Article  Google Scholar 

  4. Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.: Proximity of persistence modules and their diagrams. In: Proc. 25th, ACM Sympos. on Comput. Geom., pp. 237–246 (2009)

  5. Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules (Springer, ed.). Springer, Berlin (2016)

    Book  Google Scholar 

  6. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120 (2007)

    MathSciNet  Article  Google Scholar 

  7. Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. J. Algebra Appl. 14, 1550066, 8 pp (2014). arXiv:1210.0819

    MathSciNet  Article  Google Scholar 

  8. Curry, J.M.: Sheaves, cosheaves and applications (2013). arXiv:1303.3255v2

  9. Edelsbrunner, H., Harer, J.: Persistent homology – a survey. Surv. Discrete and Compute. Geom. Contemp. Math. 453, 257–282 (2008)

    Article  Google Scholar 

  10. Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45, 61–75 (2008)

    MathSciNet  Article  Google Scholar 

  11. Goresky, M., MacPherson, R.: Stratified Morse Theory, Ergebnisse Der Mathematik Und Ihrer Grenzgebiete, vol. 14. Springer, Berlin (1988)

    MATH  Google Scholar 

  12. Guibas, L., Ramschaw, L., Stolfi, J.: A kinetic framework for computational geometry. In: Proc. IEEE Symp. on Foundations of Computer Science, pp. 74–123 (1983)

  13. Guillermou, S.: The three cusps conjecture (2016). arXiv:1603.07876

  14. Guillermou, S., Kashiwara, M., Schapira, P.: Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems. Duke Math. J. 161, 201–245 (2012)

    MathSciNet  Article  Google Scholar 

  15. Guillermou, S., Schapira, P.: Microlocal theory of sheaves and Tamarkin’s non displaceability theorem. LN of the UMI, pp. 43–85 (2014). arXiv:1106.1576

  16. Kashiwara, M.: On the maximally overdetermined systems of linear differential equations I. Publ. Res. Inst. Math. Sci. 10, 563–579 (1975)

    MathSciNet  Article  Google Scholar 

  17. Kashiwara, M.: The Riemann–Hilbert problem for holonomic systems. Publ. RIMS Kyoto Univ. 20, 319–365 (1984)

    MathSciNet  Article  Google Scholar 

  18. Kashiwara, M., Schapira, P.: Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292. Springer, Berlin (1990)

    Google Scholar 

  19. Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Found. Comput. Math. 15, 613–650 (2015)

    MathSciNet  Article  Google Scholar 

  20. Lesnick, M., Wright, M.: Interactive visualization of 2-d persistence modules (2015). arXiv:1512.00180

  21. Oudot, S.: Persistence Theory: From Quiver Representations to Data Analysis, Mathematical Surveys and Monographs, vol. 209. AMS, Providence (2015)

    Book  Google Scholar 

  22. Schapira, P.: Operations on constructible functions. J. Pure Appl. Algebra 72, 83–93 (1991)

    MathSciNet  Article  Google Scholar 

  23. Tamarkin, D.: Microlocal conditions for non-displaceability (2008). arXiv:0809.1584

Download references


The second named author warmly thanks Gregory Ginot for having organized a seminar on persistent homology, at the origin of this paper, and Benoît Jubin for fruitful discussions on this subject. In this seminar, Nicolas Berkouk and Steve Oudot pointed out the problem of approximating constructible sheaves with objects which would be similar to higher dimensional barcodes, what we do, in some sense, here. Moreover, the links between \(\gamma \)-sheaves and persistent modules (Proposition 2.15) were clarified during discussions with Nicolas Berkouk.

Author information



Corresponding author

Correspondence to Pierre Schapira.

Additional information

The research of M.K was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science. The research of P.S was supported by the ANR-15-CE40-0007 “MICROLOCAL”.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kashiwara, M., Schapira, P. Persistent homology and microlocal sheaf theory. J Appl. and Comput. Topology 2, 83–113 (2018).

Download citation


  • Microlocal sheaf theory
  • Persistent homology
  • Barcodes

Mathematics Subject Classification

  • 55N99
  • 18A99
  • 35A27