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)
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
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|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||finite field||p-adic integers||localization at (p)||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|
In nonstandard analysis