# The Schwarz-Voronov Embedding of ${\mathbb Z}_{2}^{n}$-Manifolds

## TL;DR

This paper develops a categorical framework for ${b Z}_2^n$-manifolds, introducing the Schwarz-Voronov embedding to represent these manifolds as functors from ${b Z}_2^n$-points to Fréchet manifolds, establishing an equivalence of categories.

## Contribution

It introduces the Schwarz-Voronov embedding, providing a fully faithful functorial representation of ${b Z}_2^n$-manifolds within a functor category, and proves their categorical equivalence to locally trivial functors.

## Key findings

- Constructed a fully faithful embedding of ${b Z}_2^n$-manifolds into functor categories.
- Proved the categorical equivalence between ${b Z}_2^n$-manifolds and locally trivial functors.
- Extended the functor of points approach to ${b Z}_2^n$-graded 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 Fr\'echet 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.