nLab infinitesimal neighborhood

Infinitesimal neighbourhoods

This entry is about what in nonstandard analysis are called monads. For disambiguation see at monad (disambiguation).


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Infinitesimal neighbourhoods


Der unendlich kleinste Theil des Raumes ist immer ein Raum, etwas, das Continuität hat, nicht aber ein blosser Punct (Fichte 1795, Grundriss §4.IV)

(“An infinitesimally small part of space is always a space, something with continuity, but not a bare point.”)

An infinitesimal neighbourhood is a “neighbourhood with infinitesimal diameter”. This can be made precise in several formalisms, including nonstandard analysis, ringed spaces, synthetic differential geometry, differential cohesion, ….

“Monads”. In nonstandard analysis the infinitesimal neighbourhood of a point is traditionally called (see below) its monad, apparently following the ancient terminology going back to Euclid, for more see there.

Curiously, in differential cohesion the construction of all infinitesimal neighbourhoods (the infinitesimal disk bundle) is indeed a monad in the sense of category theory (see there, namely the left adjoint monad to the jet comonad). Whether this confluence of the terms “monads” is just a happy coincidence seems to be lost to history, see at monad the section Etymology.


In nonstandard analysis

In nonstandard analysis, the


of a standard point pp in a topological space (or even in a Choquet space) is the hyperset of all hyperpoints? infinitely close to pp. It is the (external) set constructed as the intersection of all of the standard neighbourhoods of pp, the infinitesimal neighbourhood of pp.


In differential cohesion

For H\mathbf{H} a context of differential cohesion with infinitesimal shape modality \Im, then for x:*Xx\colon \ast \to X a global point in any object XHX \in \mathbf{H} the infinitesimal disk 𝔻 x XX\mathbb{D}^X_x \to X around that point is the (homotopy) pullback of the unit i:X(X)i \colon X \to \Im(X) of the \Im-monad

𝔻 x X X i * x (X). \array{ \mathbb{D}^X_x &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{i}} \\ \ast &\stackrel{x}{\longrightarrow}& \Im(X) } \,.

The collection of all infinitesimal disks forms the infinitesimal disk bundle over XX, see there for more.

For ringed spaces

Consider a morphism (f,f ):(Y,𝒪 Y)(X,𝒪 X)(f,f^\sharp):(Y,\mathcal{O}_Y)\to(X,\mathcal{O}_X) of ringed spaces for which the corresponding map f :f *𝒪 X𝒪 Yf^\sharp:f^*\mathcal{O}_X\to\mathcal{O}_Y of sheaves on YY is surjective. Let = f=Kerf \mathcal{I} = \mathcal{I}_f = Ker\,f^\sharp, then 𝒪 Y=f (𝒪 X)/ f\mathcal{O}_Y = f^\sharp(\mathcal{O}_X)/\mathcal{I}_f. The ring f *(𝒪 Y)f^*(\mathcal{O}_Y) has the \mathcal{I}-preadic filtration which has the associated graded ring Gr = n f n/ f n+1Gr_\bullet =\oplus_{n} \mathcal{I}_f^n/\mathcal{I}^{n+1}_f which in degree 11 gives the conormal sheaf Gr 1= f/ f 2Gr_1 = \mathcal{I}_f/\mathcal{I}^2_f of ff. The 𝒪 Y\mathcal{O}_Y-augmented ringed space (Y,f (𝒪 X)/ n+1)(Y,f^\sharp(\mathcal{O}_X)/\mathcal{I}^{n+1}) is called the nn-th infinitesimal neighborhood of YY along morphism ff. Its structure sheaf is called the nn-th normal invariant of ff.

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization


In algebraic geometry (via infinitesimal shape modality):

  • A. Grothendieck, Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie, Publications Mathématiques de l’IHÉS 32 (1967), p. 5-361, numdam

Discussion of infinitesimal neighbourhoods in nonstandard analysis and use of the “monad-terminology”:

and using the “halo”-terminology:

  • Diego Rayo, Def. 4.11 in: Introduction to non-standard analysis (2015) [pdf]

See also:

More historical commentary on the monad-terminology

Discussion in synthetic differential geometry, also using the “monad”-terminology:

Discussion in differential cohesion (see at infinitesimal disk bundle):

and its formalization in differentially cohesive homotopy type theory:

Last revised on August 24, 2023 at 15:11:22. See the history of this page for a list of all contributions to it.