# nLab smooth map

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

A function which is differentiable function to arbitrary order is called a smooth function.

### In the real numbers

#### Epsilon-delta definition

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 $x \in \mathbb{R}$,

$|(D^{n}f)(x + h) - (D^{n}f)(x) - h (D^{n + 1}f)(x)| \lt \epsilon |h|$

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 $x \in \mathbb{R}$,

$\left|f(x + h) - \sum_{i=0}^n \frac{h^i (D^{i}f)(x)}{i!}\right| \lt \epsilon |h^n|$

#### Infinitesimal definition

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 $\epsilon^{n + 1} = 0$

$f_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$

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 $n \in \mathbb{N}$

$f_A(h(j(a)) + \epsilon) = \sum_{i = 0}^{\infty} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$

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 $I$.

### Between Cartesian spaces

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).

### Between smooth manifolds

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 between them.

### Between generalized smooth spaces

There are various categories of generalised smooth spaces whose morphisms are generalized smooth functions.

For details see for example at smooth set.

## Properties

Basic facts about smooth functions are

## Examples

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

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ 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 geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

## References

An early account, in the context of Cohomotopy, cobordism theory and the Pontryagin-Thom construction:

• Lev Pontrjagin, Chapter I of: Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)

Last revised on October 5, 2023 at 04:38:52. See the history of this page for a list of all contributions to it.