superpoint

**superalgebra** and (synthetic ) **supergeometry**

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.

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.

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.

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.

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.

For instance

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

Revised on March 16, 2017 07:35:04
by Urs Schreiber
(185.25.95.132)