Nori Diagrams and Persistent Homology

  • Yuri I. Manin
  • Matilde MarcolliEmail author


Recently, it was found that there is a remarkable intuitive similarity between studies in theoretical computer science dealing with large data sets on the one hand, and categorical methods of topology and geometry in pure mathematics, on the other. In this article, we treat the key notion of persistency from computer science in the algebraic geometric context involving Nori motivic constructions and related methods. We also discuss model structures for persistent topology.


Persistent homology Nori diagrams Nori motives Thin categories Model structures 

Mathematics Subject Classification

68P15 18B20 



We thank Jack Morava for suggesting the question of model structures for persistent homology discussed in Sect. 6. The second author is partially supported by NSF grant DMS-1707882, by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement Grant RGPAS-2018-522593, by the FQXi Grant FQXi-RFP-1 804, and by the Perimeter Institute for Theoretical Physics.


  1. 1.
    André, Y.: Une introduction aux motives (motifs purs, motifs mixtes, périodes.) Panoramas et Synthèses, vol. 17. Société Mathématique de France, Paris (2014) Google Scholar
  2. 2.
    Arapura, D.: An abelian category of motivic sheaves. Adv. Math. 233, 135–195 (2013). arXiv:0801.0261 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barannikov, S.A.: The Framed Morse complex and its invariants. Adv. Soviet Math. 21, 93–115 (1994)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bar-Natan, D.: On Khovanov’s categorification of the Jones polynomial. Algebr. Geom. Topol. 2, 337–370 (2002). arXiv:math.QA/0201043 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Beilinson, A., Goncharov, A., Schechtman, V., Varchenko, A.: Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane. In: The Grothendieck Festschrift in Volume I, pp. 135-172, Progress in Mathematics, Vol. 86, Birkhäuser (1990)Google Scholar
  6. 6.
    Bergner, J.: The Homotopy Theory of \((\infty, 1)\)-Categories. Cambridge University Press, Cambridge (2018)zbMATHGoogle Scholar
  7. 7.
    Blumberg, A., Lesnick, M.: Universality of the homotopy interleaving distance (2017). arXiv:1705.01690
  8. 8.
    Boissonnat, J., Chazal, F., Yvinec, M.: Geometric and Topological Inference. Cambridge University Press, Cambridge (2018)CrossRefGoogle Scholar
  9. 9.
    Brion, M.: On algebraic semigroups and monoids. In: Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics, pp. 1–54, Fields Institute Communications, vol. 71, Springer, Berlin (2014)Google Scholar
  10. 10.
    Brion, M.: On algebraic semigroups and monoids, II. Semigroup Forum 88(1), 250–272 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bubenik, P., Scott, J.: Categorification of persistent homology. Discr. Comput. Geom. 51(3), 600–627 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bubenik, P., de Silva, V., Scott, J.: Metrics for generalised persistence modules. Found. Comput. Math. 15(6), 1501–1531 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Carlsson, G.: Topology and data. Bull. Am. Math. Soc. (NS) 46(2), 255–308 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chandler, A.: Thin posets and homology theories, preprint (2018).
  15. 15.
    Dugger, D.: Universal homotopy theories. Adv. Math. 164(1), 144–176 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Edelsbrunner, H., Harer, J.: Computational Topology. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  17. 17.
    Eisenbud, D., Popescu, S., Yuzvinsky, S.: Hyperplane arrangement cohomology and monomials in the exterior algebra. Trans. Am. Math. Soc. 355(11), 4365–4383 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gelfand, S., Manin, Yu.: Methods of Homological Algebra, vol. xvii, 2nd edn, p. 372. Springer, Berlin (2003)CrossRefGoogle Scholar
  19. 19.
    Goerss, P., Jardine, R.: Simplicial Homotopy Theory. Birkhäuser, Boston (1999)CrossRefGoogle Scholar
  20. 20.
    Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. IHES 53, 53–73 (1981)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  22. 22.
    Hirschhorn, Ph: Model Categories and Their Localizations. American Mathematical Society, Providence (2003)zbMATHGoogle Scholar
  23. 23.
    Hovey, M.: Model Categories, Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1998)Google Scholar
  24. 24.
    Huber, A., Müller-Stach, St: Periods and Nori Motives. With Contributions by Benjamin Friedrich and Jonas von Wangenheim, p. xxiii+372. Springer, Berlin (2017)zbMATHGoogle Scholar
  25. 25.
    Kapranov, M., Vasserot, E.: Vertex algebras and the formal loop space. Publ. Math. IHES 100, 209–269 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kashiwara, M., Schapira, P.: Categories and Sheaves, p. x+497. Springer, Berlin (2006)CrossRefGoogle Scholar
  27. 27.
    Kashiwara, M., Schapira, P.: Persistent homology and microlocal sheaf theory, pp. 30 (2017). arXiv:1705.00955
  28. 28.
    Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000). arXiv:math.QA/9908171 MathSciNetCrossRefGoogle Scholar
  29. 29.
    Li, D.: The algebraic geometry of Harper operators. J. Phys. A 44(40), 405204 (2011)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Marcolli, M.: Gamma spaces and information. J. Geom. Phys. 140, 26–55 (2019). arXiv:1807.05314 MathSciNetCrossRefGoogle Scholar
  31. 31.
    Manin, Yu., Borisov, D.: Generalized operads and their inner cohomomorhisms . In: M. Kapranov et al. (ed.) Geometry and Dynamics of Groups and spaces (In memory of Aleksader Reznikov), Progress in Math., vol. 265. Birkhäuser, Boston, pp. 247–308 (2007). arXiv:math.CT/0609748
  32. 32.
    Mitchener, P.D.: Coarse homology theories. Algebr. Geom. Topol. 1, 271–297 (2001)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Mitsch, H.: A natural partial order for semigroups. Proc. Am. Math. Soc. 97(3), 384–388 (1986)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Murfet, D.: Abelian Categories. Preprint, (2006)
  35. 35.
    Nambooripad, K.: The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. 23, 249–260 (1980)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Nowak, P.W., Yu, G.: Large Scale Geometry, p. xiv+189. European Mathematical Society, Zurich (2012)CrossRefGoogle Scholar
  37. 37.
    Previdi, L.: Locally compact objects in exact categories. Int. J. Math. 22(12), 1787–1821 (2011)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Roe, J.: Lectures on Coarse Geometry. University Lecture Series, vol. 31, p. viii+175. American Mathematical Society, Providence (2003)zbMATHGoogle Scholar
  39. 39.
    Roe, J.: Coarse cohomology and index theory on complete Riemannian manifolds. Mem. Am. Math. Soc. 104(497), x+90 (1993)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Schwede, S., Shipley, B.: Algebras and modules in monoidal model categories. Proc. Lond. Math. Soc. (3) 80(2), 491–511 (2000). arXiv:math/9801082 MathSciNetCrossRefGoogle Scholar
  41. 41.
    Segal, G.: Categories and cohomology theories. Topology 13, 293–312 (1974)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Zomorodian, A.: Topology for Computing. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Max–Planck–Institut für MathematikBonnGermany
  2. 2.California Institute of TechnologyPasadenaUSA
  3. 3.University of TorontoTorontoCanada
  4. 4.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations