nLab formal smooth set

Idea

The concept of formal smooth set is a kind of generalized space generalizing smooth sets from plain differential geometry to differential geometry equipped with explicit infinitesimal spaces, hence to synthetic differential geometry. Just as a smooth set is equivalently a sheaf on the site of Cartesian spaces, so a formal smooth set is a sheaf on the site of formal Cartesian spaces, hence of Cartesian products of Cartesian spaces with infinitesimally thickened points. The resulting sheaf topos is also known as Dubuc’s Cahiers topos.

For the moment, for more see at geometry of physics the chapters manifolds and orbifolds and 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

In the context of jet bundles and partial differential equations:

• Igor Khavkine, Urs Schreiber: Synthetic geometry of differential equations: I. Jets and comonad structure [arXiv:1701.06238]

and further in the context of variational calculus and Lagrangian field theory:

• Grigorios Giotopoulos, Hisham Sati: Field Theory via Higher Geometry II: Thickened Smooth Sets as Synthetic Foundations [arXiv:2512.22816]