synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Der unendlich kleinste Theil des Raumes ist immer ein Raum, etwas, das Continuität hat, nicht aber ein blosser Punct, oder die Grenze zwischen bestimmten Stellen im Raume; (Fichte 1795, Grundriss §4.IV)
An infinitesimal neighbourhood is a neighbourhood with infinitesimal diameter. These can be defined in several setups: nonstandard analysis, synthetic differential geometry, ringed spaces, ….
For a context of differential cohesion with infinitesimal shape modality , then for a global point in any object the infinitesimal disk around that point is the (homotopy) pullback of the unit of the -monad
The collection of all infinitesimal disks forms the infinitesimal disk bundle over .
In nonstandard analysis, the monad or halo of a standard point in a topological space (or even in a Choquet space) is the hyperset of all hyperpoint?s infinitely close to . It is the intersection of all of the standard neighbourhoods of and is itself a hyper-neighbourhood of , the infinitesimal neighbourhood of .
It is best to avoid the term ‘monad’ for this concept on this wiki, since it has more or less nothing to with the categorial monads that are all over the place here (including elsewhere on this very page).
Consider a morphism of ringed spaces for which the corresponding map of sheaves on is surjective. Let , then . The ring has the -preadic filtration which has the associated graded ring which in degree gives the conormal sheaf of . The -augmented ringed space is called the -th infinitesimal neighborhood of along morphism . Its structure sheaf is called the -th normal invariant of .
Examples of sequences of local structures
In algebraic geometry (via infinitesimal shape modality)
Discussion in nonstandard analysis is in
Discussion in differential cohesion is in
Discussion in differentially cohesive homotopy type theory is in
Last revised on June 29, 2017 at 11:14:23. See the history of this page for a list of all contributions to it.