This entry is about what in nonstandard analysis are called monads. For disambiguation see at monad (disambiguation).
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
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
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 (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, the
or
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
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
The collection of all infinitesimal disks forms the infinitesimal disk bundle over $X$, see there for more.
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 $f$.
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 |
In algebraic geometry (via infinitesimal shape modality):
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:
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.