nLab superpoint

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 $\mathbb{R}^{0|q}$.

For more see at geometry of physics – superalgebra.

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

The category of superpoints

$$SuperPoint \hookrightarrow SuperMfd$$

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 $\mathbb{Z}/2$ graded algebras).

We have an equivalence of categories

$$SuperPoint \simeq GrAlg^{op}$$

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

The site of superpoints

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

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

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.

Related concepts

• infinitesimally thickened point

• super smooth infinity-groupoid

• super Cartan geometry

• spectral superpoint

References

For instance

• Christoph Sachse, section 2.2.1 of A Categorical Formulation of Superalgebra and Supergeometry (arXiv:0802.4067)