# nLab local diffeomorphism

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

A smooth function $f:X\to Y$ between two smooth manifolds is a local diffeomorphism if the following equivalent conditions hold

• $f$ is both a submersion and an immersion;

• for each point $x\in X$ the derivative $df:{T}_{x}X\to {T}_{f\left(x\right)}Y$ is an isomorphism of tangent vector spaces;

• the canonical diagram

$\begin{array}{ccc}TX& \stackrel{df}{\to }& TY\\ ↓& & ↓\\ X& \stackrel{f}{\to }& Y\end{array}$\array{ T X &\stackrel{d f}{\to}& T Y \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y }

(with the differential between the tangent bundles) on top is a pullback;

• for each point $x\in X$ there exists an open subset $x\in U\subset X$ such that

1. the image $f\left(U\right)$ is an open subset in $Y$;

2. $f$ restricted to $U$ is a diffeomorphism onto its image

$f{\mid }_{U}:U\stackrel{\simeq }{\to }f\left(U\right)$f|_U : U \stackrel{\simeq}{\to} f(U)

The equivalence of the conditions on tangent space with the conditions on open subsets follows by the inverse function theorem?.

## Properties

### Abstract characterization

The category SmoothMfd of smooth manifolds may naturally be thought of as sitting inside the more general context of the cohesive (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids. This is canonically equipped with a notion of infinitesimal cohesion exhibited by its inclusion into SynthDiff∞Grpd. This implies that there is an intrinsic notion of formally étale morphisms of smooth $\infty$-groupoids in general and of smooth manifolds in particular

###### Proposition

A smooth function is a formally étale morphism in this sense precisely if it is a local diffeomorphism.

See this section for more details.

Revised on March 19, 2012 11:16:59 by Urs Schreiber (89.204.155.155)