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Эта публикация цитируется в 9 научных статьях (всего в 9 статьях)
The Schwarz–Voronov Embedding of ${\mathbb Z}_{2}^{n}$-Manifolds
Andrew James Bruce, Eduardo Ibarguengoytia, Norbert Poncin Mathematics Research Unit, University of Luxembourg, Maison du Nombre 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
Аннотация:
Informally, ${\mathbb Z}_2^n$-manifolds are ‘manifolds’ with ${\mathbb Z}_2^n$-graded coordinates and a sign rule determined by the standard scalar product of their ${\mathbb Z}_2^n$-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 ${\mathbb Z}_2^n$-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ${\mathbb Z}_2^n$-points, i.e., trivial ${\mathbb Z}_2^n$-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 ${\mathbb Z}_2^n$-manifolds into a subcategory of contravariant functors from the category of ${\mathbb Z}_2^n$-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 ${\mathbb Z}_2^n$-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory.
Ключевые слова:
supergeometry, superalgebra, ringed spaces, higher grading, functor of points.
Поступила: 10 июля 2019 г.; в окончательном варианте 30 декабря 2019 г.; опубликована 8 января 2020 г.
Образец цитирования:
Andrew James Bruce, Eduardo Ibarguengoytia, Norbert Poncin, “The Schwarz–Voronov Embedding of ${\mathbb Z}_{2}^{n}$-Manifolds”, SIGMA, 16 (2020), 002, 47 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma1539 https://www.mathnet.ru/rus/sigma/v16/p2
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