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The Schwarz-Voronov embedding of \(\mathbb{Z}_2^n\)-manifolds. (English) Zbl 1433.58011

Summary: 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.

MSC:

58C50 Analysis on supermanifolds or graded manifolds
14A22 Noncommutative algebraic geometry
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