nLab
superpoint

Contents

Idea

A superpoint is an infinitesimally thickened point whose infinitesimal extension is odd in the sense of supergeometry. A super Cartesian space of vanishing ordinary dimension.

Definition

A superpoint is a supermanifold of the form 0|q\mathbb{R}^{0|q}.

The object 0|1\mathbb{R}^{0|1} is also called the odd line.

The category of superpoints

SuperPointSuperMfd SuperPoint \hookrightarrow SuperMfd

is the full subcategory of the category of supermanifolds on the superpoints.

Properties

Formal duals

The algebra of functions on superpoints are precisely the Grassmann algebras (regarded as 2\mathbb{Z}-2 graded algebras).

We have an equivalence of categories

SuperPointGrAlg op SuperPoint \simeq GrAlg^{op}

of the category of superpoints with the opposite category of Grassmann algebras.

The site of superpoints

Regard SuperPointSuperPoint as a site with trivial coverage. Much of superalgebra and supergeometry can be usefully understood as taking place over the base topos Sh(SuperPoint)Sh(SuperPoint) – the sheaf topos over superpoints – or rather the (∞,1)-sheaf (∞,1)-topos

SuperGrpd:=Sh (,1)(SuperPoint) Super\infty Grpd := Sh_{(\infty,1)}(SuperPoint)

of super ∞-groupoids. See there for more details.

Relation to super-translations and super-Minkowski spacetime

A super translation Lie algebra and hence super Minkowski spacetime, is a Lie algebra extension of a superpoint, with the latter regarded as an abelian super Lie algebra. See at super translation Lie algebra for more on this.

References

Section 2.2.1 of

  • Christoph Sachse, A Categorical Formulation of Superalgebra and Supergeometry (arXiv:0802.4067)

Revised on October 9, 2013 11:25:46 by Urs Schreiber (89.204.139.155)