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A Grothendieck topos of generalized functions I: basic theory

Paolo Giordano, Michael Kunzinger, Hans Vernaeve

Abstract

The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and sharing a number of fundamental properties with smooth functions, in particular with respect to composition and nonlinear operations. This is how they are still used in informal calculations in Physics. We introduce a category of generalized functions as smooth set-theoretical maps on (multidimensional) points of a ring of scalars containing infinitesimals and infinities. This category extends Schwartz distributions. The calculus of these generalized functions is closely related to classical analysis, with point values, composition, non-linear operations and the generalization of several classical theorems of calculus. Finally, we extend this category of generalized functions into a Grothendieck topos of sheaves over a concrete site. This topos hence provides a suitable framework for the study of spaces and functions with singularities. In this first paper, we present the basic theory; subsequent ones will be devoted to the resulting theory of ODE and PDE.

A Grothendieck topos of generalized functions I: basic theory

Abstract

The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and sharing a number of fundamental properties with smooth functions, in particular with respect to composition and nonlinear operations. This is how they are still used in informal calculations in Physics. We introduce a category of generalized functions as smooth set-theoretical maps on (multidimensional) points of a ring of scalars containing infinitesimals and infinities. This category extends Schwartz distributions. The calculus of these generalized functions is closely related to classical analysis, with point values, composition, non-linear operations and the generalization of several classical theorems of calculus. Finally, we extend this category of generalized functions into a Grothendieck topos of sheaves over a concrete site. This topos hence provides a suitable framework for the study of spaces and functions with singularities. In this first paper, we present the basic theory; subsequent ones will be devoted to the resulting theory of ODE and PDE.

Paper Structure

This paper contains 26 sections, 56 theorems, 175 equations, 3 figures.

Table of Contents

  1. Introduction: foundations of generalized functions as set-theoretical maps
  2. The ring of scalars and its topologies
  3. Topologies on $\widetilde{\mathbb{R}}^{n}$
  4. Open, closed and bounded sets generated by nets
  5. Generalized functions as smooth set-theoretical maps
  6. Definition and sharp continuity
  7. Embedding of Schwartz distributions
  8. Closure with respect to composition
  9. Differential calculus and the Fermat-Reyes theorem
  10. Integral calculus using primitives
  11. Some classical theorems for generalized smooth functions
  12. Multidimensional integration and hyperlimits
  13. Integration over functionally compact sets
  14. Hyperfinite limits
  15. Properties of multidimensional integral
  16. ...and 11 more sections

Key Result

Theorem 2

$\widetilde{\mathbb{R}}$ is a partially ordered ring. The real numbers $r\in\mathbb{R}$ are embedded in $\widetilde{\mathbb{R}}$ by viewing them as constant nets $[r]\in\widetilde{\mathbb{R}}$.

Figures (3)

  • Figure 3.1: A representation of Dirac delta and Heaviside function. A Colombeau mollifier has a representation similar to Dirac delta (but with finite values).
  • Figure 3.2: A representation of $\delta\circ\delta$
  • Figure 6.1: A net $(f_{\varepsilon})$ defining a discontinuous solution of a smooth equation.

Theorems & Definitions (151)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • Lemma 8
  • ...and 141 more