nLab infinitesimal neighborhood

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

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Idea

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.

Definition

In nonstandard analysis

In nonstandard analysis, the

• “monad” (Robinson 1966, p. 57 following , see also Luxemburg 1966, Keisler 1976, Def. 1.2, Kutateladze 2011)

or

• “halo” (eg. Rayo 2015, Def. 4.11)

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

Some

In differential cohesion

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

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

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

For ringed spaces

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

Related concepts

• jet groupoid

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

References

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”:

• Abraham Robinson, p. 57 of: Non-standard analysis, Studies in Logic and the Foundations of Mathematics 42, North-Holland (1966), Princeton University Press (1996) [ISBN:9780691044903]

• Wilhemus A. J. Luxemburg, A General Theory of Monads, in: Applications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Minston (1966) 18–86 [ark]

• E. I. Gordon, A. G. Kusraev, Semën S. Kutateladze, Chapter 4 in: Infinitesimal Analysis, Mathematics and its Applications 544, Springer (2002) [doi:10.1007/978-94-017-0063-4]

• Jerome Keisler, Def. 1.2 in: Foundations of Infinitesimal Calculus, Prindle Weber & Schmidt (1976, 2022) [pdf]

• Semën S. Kutateladze, Leibnizian, Robinsonian, and Boolean valued monads, Journal of Applied and Industrial Mathematics 5 3 (2011) 365-373 [arxiv/1106.2755, doi:10.1134/S1990478911030082]

• Sergio Albeverio, Jens Erik Fenstad, Raphael Hoegh-Krohn, Nonstandard methods in stochastic analysis and mathematical physics, Academic Press 1986

and using the “halo”-terminology:

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

See also:

• Wikipedia, Monad (non-standard analysis)

More historical commentary on the monad-terminology

• Semën S. Kutateladze, Leibniz’s Definition of Monad, NeuroQuantology 4 3 (2006) 249-241 [arXiv:math/0608298, pdf]

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

• Anders Kock, Formal manifolds and synthetic theory of jet bundles, Cahiers de Topologie et Géométrie Différentielle Catégoriques 21 3 (1980) [numdam:CTGDC_1980__21_3_227_0]

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

• Igor Khavkine, Urs Schreiber, Synthetic geometry of differential equations: I. Jets and comonad structure (arXiv:1701.06238)

and its formalization in differentially cohesive homotopy type theory:

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory (2017)