nLab super smooth set

Idea

Just as a smooth set is a generalized smooth space in differential geometry modeled as a sheaf on the category of Cartesian spaces with smooth functions between them, so a super smooth set is a sheaf on the category of super Cartesian spaces, being a generalized space in supergeometry.

Since super smooth sets contain infinitesimal spaces, it makes good sense to make this explicit and consider super formal smooth sets right away, hence sheaves on super formal Cartesian spaces

For details see at geometry of physics – supergeometry.

Related concepts

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$	$\phantom{A}$site$\phantom{A}$	$\phantom{A}$sheaf topos$\phantom{A}$	$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$	$\phantom{A}$Point$\phantom{A}$	$\phantom{A}$Set$\phantom{A}$	$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$CartSp$\phantom{A}$	$\phantom{A}$SmoothSet$\phantom{A}$	$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$	$\phantom{A}$FormalCartSp$\phantom{A}$	$\phantom{A}$FormalSmoothSet$\phantom{A}$	$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$SuperFormalCartSp$\phantom{A}$	$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$	$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

References

• Urs Schreiber: §4.6 in: Differential Cohomology in a Cohesive ∞-Topos

• Urs Schreiber: §3.1 in: Higher Prequantum Geometry, chapter in: New Spaces for Mathematics and Physics, Cambridge University Press (2021) [doi:10.1017/9781108854399.008, arXiv:1601.05956]

• Hisham Sati, Urs Schreiber, §3.1.3 in: Proper Orbifold Cohomology [arXiv:2008.01101]

• Grigorios Giotopoulos: Sheaf Topos Theory as a setting for Physics, talk at Workshop on Noncommutative and Generalized Geometry in String Theory, Corfu Summer Institute (2024) [pdf]