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)
A function which is differentiable function to arbitrary order is called a smooth function.
Let $\mathbb{R}$ be the real numbers. A function $f:\mathbb{R} \to \mathbb{R}$ is smooth if it comes with a sequence of functions $D^{(-)}f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$ and a sequence of functions $M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+)$ in the positive rational numbers, such that
for every real number $x \in \mathbb{R}$, $(D^{0}f)(x) = f(x)$
for every natural number $n \in \mathbb{N}$, for every positive rational number $\epsilon \in \mathbb{Q}_+$, for every real number $h \in \mathbb{R}$ such that $0 \lt | h | \lt M^{n}f(\epsilon)$, and for every real number $x \in \mathbb{R}$,
Unwrapping the recursive definition above, a function $f:\mathbb{R} \to \mathbb{R}$ is smooth if it comes with a sequence of functions $D^{(-)}f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$ and a sequence of functions $M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+)$ in the positive rational numbers, such that
for every real number $x \in \mathbb{R}$, $(D^{0}f)(x) = f(x)$
for every natural number $n \in \mathbb{N}$, for every positive rational number $\epsilon \in \mathbb{Q}_+$, for every real number $h \in \mathbb{R}$ such that $0 \lt | h | \lt M^{n}f(\epsilon)$, and for every real number $x \in \mathbb{R}$,
Given a predicate $P$ on the real numbers $\mathbb{R}$, let $I$ denote the set of all elements in $\mathbb{R}$ for which $P$ holds. A partial function $f:\mathbb{R} \to \mathbb{R}$ is equivalently a function $f:I \to \mathbb{R}$ for any such predicate $P$ and set $I$.
A function $f:I \to \mathbb{R}$ is smooth at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $\frac{d^0 f}{d x^0}\left(a\right) = a$ for all $a \in S$, such that for all Archimedean ordered Artinian local $\mathbb{R}$-algebras $A$ with ring homomorphism $h_A:\mathbb{R} \to A$ and nilradical $D$, natural numbers $n \in \mathbb{N}$, and purely infinitesimal elements $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$
Equivalently, let $\mathbb{R}[[\epsilon]]$ denote the ring of univariate formal power series on $\mathbb{R}$. $\mathbb{R}[[\epsilon]]$ is a Artinian local $\mathbb{R}$-algebra with homomorphism $h:\mathbb{R} \to \mathbb{R}[[\epsilon]]$. A function $f:I \to \mathbb{R}$ is smooth at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $\frac{d^0 f}{d x^0}\left(a\right) = a$ for all $a \in S$, such that for all natural numbers $n \in \mathbb{N}$
A function $f:I \to \mathbb{R}$ is smooth at an element $a \in I$ if it is smooth at the singleton subset $\{a\}$, and a function $f:I \to \mathbb{R}$ is smooth if it is smooth at the improper subset of $I$.
A function on (some open subset of) a cartesian space $\mathbb{R}^n$ with values in the real line $\mathbb{R}$ is smooth, or infinitely differentiable, if all its derivatives exist at all points. More generally, if $A \subseteq \mathbb{R}^n$ is any subset, a function $f: A \to \mathbb{R}$ is defined to be smooth if it has a smooth extension to an open subset containing $A$.
By coinduction: A function $f : \mathbb{R} \to \mathbb{R}$ is smooth if (1) its derivative exists and (2) the derivative is itself a smooth function.
For $A \subseteq \mathbb{R}^n$, a smooth map $\phi: A \to \mathbb{R}^m$ is a function such that $\pi \circ \phi$ is a smooth function for every linear functional $\pi: \mathbb{R}^m \to \mathbb{R}$. (In the case of finite-dimensional codomains as here, it suffices to take the $\pi$ to range over the $m$ coordinate projections.)
The concept can be generalised from cartesian spaces to Banach spaces and some other infinite-dimensional spaces. There is a locale-based analogue suitable for constructive mathematics which is not described as a function of points but as a special case of a continuous map (in the localic sense).
A topological manifold whose transition functions are smooth maps is a smooth manifold. A smooth function between smooth manifolds is a function that (co-)restricts to a smooth function between subsets of Cartesian spaces, as above, with respect to any choice of atlases, hence which is a $k$-fold differentiable function (see there for more details), for all $k$ The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth maps betweeen them.
There are various categories of generalised smooth spaces whose morphisms are generalized smooth functions.
For details see for example at smooth set.
Basic facts about smooth functions are
Every analytic functions (for instance a holomorphic function) is also a smooth function.
A crucial property of smooth functions, however, is that they contain also bump functions.
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 |
An early account, in the context of Cohomotopy, cobordism theory and the Pontryagin-Thom construction:
Last revised on June 18, 2023 at 22:22:05. See the history of this page for a list of all contributions to it.