The Schwarz-Voronov Embedding of ${\mathbb{Z}}_2^n$-Manifolds

Abstract

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.

References in corpus (15)

• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie Symmetries of the Lévy-Leblond Equations
• On a ${\mathbb{Z}}_2^n$-Graded Version of Supersymmetry
• Higher Trace and Berezinian of Matrices over a Clifford Algebra
• Splitting theorem for ${\mathbb{Z}}_2^n$-supermanifolds
• Introduction to graded geometry
• Super-de Sitter and alternative super-Poincaré symmetries
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ generalizations of $\mathcal{N}=2$ super Schrödinger algebras and their representations
• Global Analytic Approach to Super Teichmueller Spaces
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ generalizations of $\mathcal{N}=1$ superconformal Galilei algebras and their representations
• Homotopical Algebraic Context over Differential Operators
• Differential calculus on ${\mathbb{Z}}_2^n$-supermanifolds
• The Frobenius theorem for ${\mathbb{Z}}_2^n$-supermanifolds
• Infinite-Dimensional Supermanifolds via Multilinear Bundles
• The Gelfand-Naimark-Segal construction for unitary representatins of ${\mathbb{Z}}_2^n$-graded Lie supergroups
• The local functors of points of Supermanifolds

Cited by in corpus (7)

• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded supersymmetry: $2$-d sigma models
• Irreducible representations of ${\mathbb{Z}}_2^2$-graded $\mathcal{N}=2$ supersymmetry algebra and ${\mathbb{Z}}_2^2$-graded supermechanics
• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
• Local Forms of Morphisms of Colored Supermanifolds
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• Symplectic ${\mathbb{Z}}_2^n$-manifolds
• Linear ${\mathbb{Z}}_2^n$-Manifolds and Linear Actions