# nLab local diffeomorphism

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

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Definition

###### 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 $d f : T_x X \to T_{f(x)} Y$ is an isomorphism of tangent vector spaces;

• the canonical diagram

$\array{ T X &\stackrel{d f}{\to}& T Y \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y }$

(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 $f(U)$ is an open subset in $Y$;

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

$f|_U : U \stackrel{\simeq}{\to} f(U)$

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

###### Remark

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

## 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 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.

Discussion in the synthetic differential geometry of the Cahiers topos is in