The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds
Informally, ℤⁿ₂-manifolds are 'manifolds' with ℤⁿ₂-graded coordinates and a sign rule determined by the standard scalar product of their ℤⁿ₂-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in i...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
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| Sprache: | English |
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Інститут математики НАН України
2020
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Bruce, Andrew James Ibarguengoytia, Eduardo Poncin, Norbert 2025-12-12T10:41:28Z 2020 The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds. Andrew James Bruce, Eduardo Ibarguengoytia and Norbert Poncin. SIGMA 16 (2020), 002, 47 pages 1815-0659 2010 Mathematics Subject Classification: 58C50; 58D1; 14A22 arXiv:1906.09834 https://nasplib.isofts.kiev.ua/handle/123456789/210608 https://doi.org/10.3842/SIGMA.2020.002 Informally, ℤⁿ₂-manifolds are 'manifolds' with ℤⁿ₂-graded coordinates and a sign rule determined by the standard scalar product of their ℤⁿ₂-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ℤⁿ₂-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ℤⁿ₂-points, i.e., trivial ℤⁿ₂-manifolds for which the reduced manifold is just a single point, as 'probes' when employing the functor of points. This allows us to construct a fully faithful restricted Yoneda embedding of the category of ℤⁿ₂-manifolds into a subcategory of contravariant functors from the category of ℤⁿ₂-points to a category of Fréchet manifolds over algebras. We refer to this embedding as the Schwarz-Voronov embedding. We further prove that the category of ℤⁿ₂-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory. The authors cordially thank the anonymous referees for their valuable remarks and comments, which have served to improve this article, as well as for their suggestions for future research. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| spellingShingle |
The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds Bruce, Andrew James Ibarguengoytia, Eduardo Poncin, Norbert |
| title_short |
The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| title_full |
The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| title_fullStr |
The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| title_full_unstemmed |
The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| title_sort |
schwarz-voronov embedding of ℤⁿ₂-manifolds |
| author |
Bruce, Andrew James Ibarguengoytia, Eduardo Poncin, Norbert |
| author_facet |
Bruce, Andrew James Ibarguengoytia, Eduardo Poncin, Norbert |
| publishDate |
2020 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Informally, ℤⁿ₂-manifolds are 'manifolds' with ℤⁿ₂-graded coordinates and a sign rule determined by the standard scalar product of their ℤⁿ₂-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ℤⁿ₂-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ℤⁿ₂-points, i.e., trivial ℤⁿ₂-manifolds for which the reduced manifold is just a single point, as 'probes' when employing the functor of points. This allows us to construct a fully faithful restricted Yoneda embedding of the category of ℤⁿ₂-manifolds into a subcategory of contravariant functors from the category of ℤⁿ₂-points to a category of Fréchet manifolds over algebras. We refer to this embedding as the Schwarz-Voronov embedding. We further prove that the category of ℤⁿ₂-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210608 |
| citation_txt |
The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds. Andrew James Bruce, Eduardo Ibarguengoytia and Norbert Poncin. SIGMA 16 (2020), 002, 47 pages |
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2025-12-17T12:04:19Z |
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2025-12-17T12:04:19Z |
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