superalgebra

and

supergeometry

# Contents

## 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

$\mathrm{SuperPoint}↪\mathrm{SuperMfd}$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 $ℤ-2$ 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 $\mathrm{Sh}\left(\mathrm{SuperPoint}\right)$ – the sheaf topos over superpoints – or rather the (∞,1)-sheaf (∞,1)-topos

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

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

## References

Section 2.2.1 of

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

Revised on March 23, 2011 22:35:48 by Urs Schreiber (82.113.99.20)