Functor of points of a supermanifold

Definition

For a supermanifold $X$ , its functor of points is the contravariant functor

h_X:\mathbf{SMan}_{\mathbb R}^{\mathrm{sm},\mathrm{op}} \longrightarrow\mathbf{Set}, \qquad h_X(S)=\operatorname{Hom}_{\mathbf{SMan}}(S,X).

An element of $h_X(S)$ is an $S$ -point of $X$ , or equivalently a family of points of $X$ parametrized by the test supermanifold $S$ . A morphism $X\to Y$ induces a natural transformation $h_X\to h_Y$ .

By the Yoneda lemma, $X\mapsto h_X$ is fully faithful. Consequently, a supermanifold and every one of its morphisms can be characterized by all $S$ -points and their naturality in $S$ .

Why generalized points matter

Maps from the ordinary one-point manifold detect only the reduced points $|X|$ ; odd coordinates vanish there. Maps from test objects with odd nilpotents, such as purely odd affine supermanifolds, detect the odd and nilpotent directions. Coordinate formulas used in physics become ordinary formulas with Grassmann-valued coefficients after evaluation on suitable test objects.

Restricting tests to finite Grassmann algebras can be effective, but a bare collection of sets of Grassmann-valued points is not automatically equivalent to a supermanifold. One must retain naturality and, depending on the chosen formalism, the smooth or enriched structure on those point sets.

References

P. Deligne and J. W. Morgan, “Notes on supersymmetry (following Joseph Bernstein),” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999, 41–97. Relevant: functor-of-points conventions.
C. Carmeli, L. Caston, and R. Fioresi, Mathematical Foundations of Supersymmetry, EMS, 2011. Publisher record. Relevant: Chapters 4 and 10.