id ⊣ id ∨ ∨ fermionic ⇉ ⊣ ⇝ bosonic ⊥ ⊥ bosonic ⇝ ⊣ Rh rheonomic ∨ ∨ reduced ℜ ⊣ ℑ infinitesimal ⊥ ⊥ infinitesimal ℑ ⊣ & étale ∨ ∨ cohesive esh ⊣ ♭ 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 }

Models

Models for Smooth Infinitesimal Analysis
- smooth algebra (C ∞C^\infty-ring)
- smooth locus
- Fermat theory
Cahiers topos
smooth ∞-groupoid
formal smooth ∞-groupoid
super formal smooth ∞-groupoid

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Definition

A function on (some open subset of) a cartesian space ℝ n\mathbb{R}^n with values in the real line ℝ\mathbb{R} is smooth, or infinitely differentiable, if all its derivatives exist at all points. More generally, if A⊆ℝ nA \subseteq \mathbb{R}^n is any subset, a function f:A→ℝf: A \to \mathbb{R} is defined to be smooth if it has a smooth extension to an open subset containing AA.

By coinduction: A function f:ℝ→ℝf : \mathbb{R} \to \mathbb{R} is smooth if (1) its derivative exists and (2) the derivative is itself a smooth function.

For A⊆ℝ nA \subseteq \mathbb{R}^n, a smooth map ϕ:A→ℝ m\phi: A \to \mathbb{R}^m is a function such that π∘ϕ\pi \circ \phi is a smooth function for every linear functional π:ℝ m→ℝ\pi: \mathbb{R}^m \to \mathbb{R}. (In the case of finite-dimensional codomains as here, it suffices to take the π\pi to range over the mm 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 manifold whose transition functions are smooth maps is a smooth manifold. The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth maps betweeen them.

Yet more generally, the morphisms between generalised smooth spaces are smooth maps.

For functions between manifolds that fall short of full smoothness, see differentiable map.

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

geometry	point	first order infinitesimal	⊂\subset	formal = arbitrary order infinitesimal	⊂\subset	local = stalkwise	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	𝔽 p\mathbb{F}_p finite field			ℤ p\mathbb{Z}_p p-adic integers		ℤ (p)\mathbb{Z}_{(p)} localization at (p)	ℤ\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

Revision on August 13, 2017 at 14:01:03 by Urs Schreiber See the history of this page for a list of all contributions to it.

smooth map (Rev #18) in nLab

Context

Differential geometry

Contents

Definition

Properties

Examples