A Grothendieck topos of generalized functions I: basic theory
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| Format: | Preprint |
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2021
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| _version_ | 1866917762755985408 |
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| author | Giordano, Paolo Kunzinger, Michael Vernaeve, Hans |
| author_facet | Giordano, Paolo Kunzinger, Michael Vernaeve, Hans |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_04492 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Grothendieck topos of generalized functions I: basic theory Giordano, Paolo Kunzinger, Michael Vernaeve, Hans Functional Analysis 46-XX, 46Fxx, 30Gxx, 46Txx, 46F30, 26E30, 58A03 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. |
| title | A Grothendieck topos of generalized functions I: basic theory |
| topic | Functional Analysis 46-XX, 46Fxx, 30Gxx, 46Txx, 46F30, 26E30, 58A03 |
| url | https://arxiv.org/abs/2101.04492 |