Equivalence functors in graded supergeometry

TL;DR

This paper explores the categorical equivalences between N-manifolds of degree n and n-tuple vector superbundles, providing a geometric framework that generalizes and clarifies the desuperization process using standard supergeometry tools.

Contribution

It offers a broad, canonical geometric framework for understanding equivalences between graded supermanifolds and vector superbundles, extending previous results on N-manifolds.

Findings

Explicit description of categorical equivalences

Construction of desuperization functor as a composition of canonical maps

Use of standard supergeometry tools for constructions

Abstract

It has recently been proved that the category of N-manifolds of degree $n$, that is, $\mathbb N$-graded supermanifolds of degree $n$ for which the parity agrees with the gradation, is equivalent to the category of purely even $n$-tuple vector superbundles equipped with a suitable action of the symmetric group $S_n$ permuting the vector bundle structures. This equivalence may be interpreted as a `desuperization' of N-manifolds. In the present paper, we place this result within a broader framework of graded structures on supermanifolds and explicitly describe several canonical equivalences between the corresponding categories in a purely geometric, constructive manner. The desuperization equivalence functor appears as a composition of some of these canonical equivalences. Our constructions are entirely canonical and rely on standard tools of supergeometry, including iterated tangent functors, parity reversion in vector superbundles, and the interpretation of $n$-tuple vector bundles in terms of commuting Euler vector fields associated with the underlying vector bundle structures.