synthetic differential geometry, deformation theory
infinitesimally thickened point
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A formal neighbourhood is a neighbourhood in formal geometry, hence an “infinitesimal neighbourhood”. Also called a formal completion or formal disk, see there for more.
The formal spectrum of the completion of a commutative ring in the $I$-adic topology for a maximal ideal $I$ is the formal neighbourhood of the point corresponding to that ideal in the spectrum of a commutative ring.
perturbative quantum field theory (in particular realized as formal deformation quantization) describes the infinitesimal neighbourhood of classical free field theory in the space of all quantum field theories.
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |