On the definition of superspace

The Schwarz-Voronov Embedding of Z₂ⁿ-Manifolds

“…, γ N , and with coefficients in arbitrary R-valued functions on U p , i.e., we do not ask that these functions be continuous let alone smooth. Following [49,54,58], we will refer to this algebra as a Z n 2 -Berezin algebra. Any element of this algebra is of the form…”

“…To quote Schmitt [47]: "However, no one has ever measured a Grassmann number, everyone measures real numbers". The solution here is, following Schwarz & Voronov [49,54,58], not to fix the underlying Grassmann algebra, but rather understand supermanifolds as functors from the category of finite-dimensional Grassmann algebras to, in the first instance, the category of sets. For a given, but arbitrary, Grassmann algebra Λ, one speaks of the set of Λ-points of a supermanifold.…”

“…Dual to Z n 2 -points are what we will call Z n 2 -Grassmann algebras, which we will also denote as Λ. The aim of this paper is to generalize and carefully prove the main results of Schwarz & Voronov [54,58] to the 'higher graded' setting. In particular, we show that Z n 2 -points are sufficient to act as 'probes' when employing the functor of points.…”

On curves and jets of curves on supermanifolds

“…Remark Following the work of Schwarz and Voronov [17,20] it is known that it is actually sufficient to consider just Λ-points, that is supermanifolds of the form R 0|q (q ≥ 1) as paramaterisations.…”

Automorphism groups of N=2 superconformal super-Riemann spheres

“…However within the concrete approach (in particular allowing for odd functions of an even variable), certain functorial properties can be incorporated into the definition of superfunction and supermanifold alleviating the need for a particular choice of underlying Grassmann algebra (cf. [25]); or even further generalizations can be made by replacing the underlying Grassmann algebras with almost nilpotent superalgebras [19]. Although we restrict most of our discussion in this paper to supermanifolds over Grassmann algebras, the extension to almost nilpotent algebras following [19] is a natural one, as is the functorial nature of our definitions; see Remarks 2.1 and 2.3.…”