Products in the category of ${\mathbb{Z}}_2^n$-manifolds

Abstract

We prove that the category of ${\mathbb{Z}}_2^n$-manifolds has all finite products. Further, we show that a ${\mathbb{Z}}_2^n$-manifold (resp., a ${\mathbb{Z}}_2^n$-morphism) can be reconstructed from its algebra of global ${\mathbb{Z}}_2^n$-functions (resp., from its algebra morphism between global ${\mathbb{Z}}_2^n$-function algebras). These results are of importance in the study of ${\mathbb{Z}}_2^n$ Lie groups. The investigation is all the more challenging, since the completed tensor product of the structure sheafs of two ${\mathbb{Z}}_2^n$-manifolds is not a sheaf. We rely on a number of results on (pre)sheaves of topological algebras, which we establish in the appendix.

arXiv comment38 pages. Comments welcomed

References in corpus (3)

• Higher Trace and Berezinian of Matrices over a Clifford Algebra
• Splitting theorem for ${\mathbb{Z}}_2^n$-supermanifolds
• Supermetrics on supermanifolds

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• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ generalizations of infinite dimensional Lie superalgebra of conformal type with complete classification of central extensions
• Symplectic ${\mathbb{Z}}_2^n$-manifolds
• Principal bundles in the category of ${\mathbb{Z}}_2^n$-manifolds