nLab superpoint






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

For more see at geometry of physics – superalgebra.

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.


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.


For instance

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

Last revised on March 16, 2017 at 11:35:04. See the history of this page for a list of all contributions to it.