nLab
superpoint

Idea

A superpoint is an infinitesimally thickened point whose infinitesimal extension is odd in the sense of supergeometry.

Definition

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

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

The category of superpoints

$$\mathrm{SuperPoint}\hookrightarrow \mathrm{SuperMfd}$$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

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

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

The site of superpoints

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

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

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

Related concepts

• infinitesimally thickened point

References

Section 2.2.1 of

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