and
Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object in there.
As ordinary differential geometry studies spaces – smooth manifolds – that locally look like vector spaces, supergeometry studies spaces – supermanifolds – that locally look like super vector spaces.
As ordinary algebraic geometry studies spaces – schemes – that locally look like affine spaces, supergeometry studies superscheme?s.
From the point of view of noncommutative geometry, the supergeometry is a very mild special case of noncommutativity in geometry: some coordinates commute, some anticommute.
Some influential general considerations are in
The formulation of supergeometry as geometry over the topos over the category of superpoints is reviewed in
For many more references see at supermanifold.
Plenty of discussion of supergeometry with an eye towards supersymmetry in quantum field theory is in
especially in the contribution
The appendix there
means to sort out various sign conventions of relevance.
Discussion of how supersymmetry is universally induced in higher category theory/homotopy theory by the free abelian ∞-group on the point – the sphere spectrum – is in
For more on this see at superalgebra.