Arabian Journal of Mathematics

, Volume 4, Issue 4, pp 301–328

Categorical frameworks for generalized functions

Open Access
Article

Abstract

We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Frölicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frölicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frölicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Frölicher spaces) as a framework for Schwartz distributions, and prove that in the category of diffeological spaces, both the special and the full Colombeau algebras are smooth differential algebras, with a smooth embedding of Schwartz distributions and smooth pointwise evaluations of Colombeau generalized functions.

Mathematics Subject Classification

46T30 46F25 46F30 58Dxx 

References

  1. 1.
    Albeverio S., Gielerak R., Russo F.: A two-space dimensional semilinear heat equation perturbed by (Gaussian) white noise. Probab. Theory Relat. Fields 121(3), 319–366 (2001)MATHCrossRefGoogle Scholar
  2. 2.
    Adamek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories: The Joy of Cats. Wiley, New York (1990)Google Scholar
  3. 3.
    Aragona J., Juriaans S.O., Oliveira O.R.B., Scarpalézos D.: Algebraic and geometric theory of the topological ring of Colombeau generalized functions. Proc. Edinb. Math. Soc. (2) 51(3), 545–564 (2008)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Baez J.C., Hoffnung A.E.: Convenient categories of smooth spaces. Trans. Am. Math. Soc. 363(11), 5789–5825 (2011)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Batubenge, A., Iglesias-Zemmour, P., Karshon, Y., Watts, J.: Diffeological, Frölicher, and differential spaces (preprint). http://www.math.uiuc.edu/~jawatts/papers/reflexive
  6. 6.
    Boman J.: Differentiability of a function and of its compositions with functions of one variable. Math. Scand. 20, 249–268 (1967)MATHMathSciNetGoogle Scholar
  7. 7.
    Burtscher A., Kunzinger M.: Algebras of generalized functions with smooth parameter dependence. Proc. Edinb. Math. Soc., (2) 55(1), 105–124 (2012)MATHMathSciNetGoogle Scholar
  8. 8.
    Christensen J.D., Sinnamon G., Wu E.: The D-topology for diffeological spaces. Pac. J. Math. 272(1), 87–110 (2014)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Christensen, J.D., Wu, E.: Tangent spaces and tangent bundles for diffeological spaces (prepreint). http://arxiv.org/abs/1411.5425
  10. 10.
    Colombeau, J.F.: New Generalized Functions and Multiplication of Distributions. Notas de Matemática, vol. 90. North-Holland, Amsterdam (1984)Google Scholar
  11. 11.
    Colombeau, J.F.: Elementary Introduction to New Generalized Functions. North-Holland Mathematics Studies, vol. 113. North-Holland, Amsterdam (1985)Google Scholar
  12. 12.
    Colombeau, J.F.: Multiplication of Distributions: A Tool in Mathematics, Numerical Engineering and Theoretical Physics. Lecture Notes in Mathematics, vol. 1532. Springer, Berlin (1992)Google Scholar
  13. 13.
    Frölicher, A., Kriegl, A.: Linear Spaces and Differentiation Theory. Wiley, Chichester (1988)Google Scholar
  14. 14.
    Garetto, C.: Topological structures in Colombeau algebras: topological \({{\widetilde{\mathbb{C} }}}\) -modules and duality theory. Acta Appl. Math. 88(1), 81–123 (2005)Google Scholar
  15. 15.
    Garetto, C.: Topological structures in Colombeau algebras: investigation of the duals of \({{{\mathcal{G}}_c(\Omega),{\mathcal{G}}(\Omega)}}\) and \({{{\mathcal{G}}_{\mathcal{S}}(\mathbb{R}^n)}}\) . Monatsh. Math. 146(3), 203–226 (2005)Google Scholar
  16. 16.
    Giordano, P.: Fermat reals: nilpotent infinitesimals and infinite dimensional spaces (preprint). http://arxiv.org/abs/0907.1872
  17. 17.
    Giordano P.: The ring of Fermat reals. Adv. Math. 225(4), 2050–2075 (2010)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Giordano P.: Infinitesimals without logic. Russ. J. Math. Phys. 17(2), 159–191 (2010)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Giordano P.: Infinite dimensional spaces and Cartesian closedness. Zh. Mat. Fiz. Anal. Geom. 7(3), 225–284 (2011)MATHMathSciNetGoogle Scholar
  20. 20.
    Giordano P.: Fermat–Reyes method in the ring of Fermat reals. Adv. Math. 228(2), 862–893 (2011)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Giordano, P., Kunzinger, M.: Generalized functions as a category of smooth set-theoretical maps. http://www.mat.univie.ac.at/~giordap7/GenFunMaps
  22. 22.
    Giordano P., Kunzinger M.: New topologies on Colombeau generalized numbers and the Fermat–Reyes theorem. J. Math. Anal. Appl. 399(1), 229–238 (2013)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Giordano P., Kunzinger M.: Topological and algebraic structures on the ring of Fermat reals. Isr. J. Math. 193(1), 459–505 (2013)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions with Applications to General Relativity. Mathematics and Its Applications, vol. 537. Kluwer, Dordrecht (2001)Google Scholar
  25. 25.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, I: Distribution Theory and Fourier Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 256. Springer, Berlin (1983)Google Scholar
  26. 26.
    Iglesias-Zemmour, P.: Diffeology. Mathematical Surveys and Monographs, vol. 185. American Mathematical Society, Providence (2013)Google Scholar
  27. 27.
    Kelley, J.L., Namioka, I.: Linear Topological Spaces. Graduate Texts in Mathematics, vol. 36. Springer, New York (1976)Google Scholar
  28. 28.
    Kock A., Reyes G.: Distributions and heat equation in SDG. Cah. Topol. Géom. Différ. Catég. 47(1), 2–28 (2006)MATHMathSciNetGoogle Scholar
  29. 29.
    Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)Google Scholar
  30. 30.
    Laubinger, M.: Differential geometry in Cartesian closed categories of smooth spaces. Ph.D. thesis, Louisiana State University. http://etd.lsu.edu/docs/available/etd-02212008-165645/
  31. 31.
    Moerdijk, I., Reyes, G.E.: Models for Smooth Infinitesimal Analysis. Springer, New York (1991)Google Scholar
  32. 32.
    nLab. Topological notions of Frölicher spaces. See http://ncatlab.org/nlab/show/topological+notions+of+Fr%C3%B6licher+spaces
  33. 33.
    Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Research Notes in Mathematics Series, vol. 259. Longman, Harlow (1992)Google Scholar
  34. 34.
    Oberguggenberger M.: Generalized functions in nonlinear models—a survey. Nonlinear Anal. 47(8), 5029–5040 (2001)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Oberguggenberger, M., Russo, F.: Singular limiting behavior in nonlinear stochastic wave equations. In: Stochastic Analysis and Mathematical Physics. Progress in Probability, vol. 50, pp. 87–99. Birkhäuser, Boston (2001)Google Scholar
  36. 36.
    Schwartz, L.: Théorie des distributions, Tome I & II. Actualités Sci. Ind., vol. 1091/1122. Hermann & Cie, Paris (1950/1951)Google Scholar
  37. 37.
    Schwartz L.: Sur l’impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239, 847–848 (1954)MATHMathSciNetGoogle Scholar
  38. 38.
    Stacey A.: Comparative smootheology. Theory Appl. Categ. 25(4), 64–117 (2011)MATHMathSciNetGoogle Scholar
  39. 39.
    Steinbauer R., Vickers J.A.: On the Geroch–Traschen class of metrics. Class. Quantum Gravity 26(6), 1–19 (2009)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Shimakawa, K., Yoshida, K., Haraguchi, T.: Homology and cohomology via enriched bifunctors (preprint). http://arxiv.org/abs/1010.3336
  41. 41.
    Wu, E.: A homotopy theory for diffeological spaces. Ph.D. thesis, Western University (2012)Google Scholar
  42. 42.
    Wu, E.: Homological algebra for diffeological vector spaces. Homol. Homotopy Appl. (prepreint). http://arxiv.org/abs/1406.6717

Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.University of ViennaViennaAustria

Personalised recommendations