# nLab smooth map

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

A function on (some subset of) a cartesian space ${ℝ}^{n}$ with values in the real line $ℝ$ is smooth, or infinitely differentiable, if all its derivatives exist at all points.

By coinduction: A function $f:ℝ\to ℝ$ is smooth if 1. it’s derivative exists and 2. the derivative is itself a smooth function.

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 functions is a smooth manifold. The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth functions 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 infinitesimal and local structures

first order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$←$ differentiationintegration $\to$
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Revised on February 6, 2013 18:35:45 by Urs Schreiber (82.113.106.234)