nLab
local diffeomorphism

Context

Étale morphisms

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

Definition

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

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

Remark

An analogous characterization of étale morphisms between affine algebraic varieties isgiven by tangent cones. See there.

Properties

General

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 differential 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 November 22, 2013 02:09:42 by Urs Schreiber (77.251.114.72)