In nonstandard analysis, the monad 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 .
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|