# nLab Banach manifold

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

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

A notion of infinite-dimensional manifold. A Banach manifold is a manifold modelled on Banach spaces. By default, transition maps are taken to be smooth.

## Properties

### As absolute neighbourhood retracts

###### Example

Every paracompact Banach manifold is an absolute neighbourhood retract.

By Palais 1966, Cor. to Thm. 5 on p. 3.

### Embedding into the category of diffeological spaces

The category of smooth Banach manifolds has a full and faithful functor into the category of diffeological spaces. In terms of Chen smooth spaces this was observed in (Hain). For more see at Fréchet manifold – Relation to diffeological spaces.

## References

For general references see at infinite-dimensional manifold.

Aspects of the homotopy theory of Banach manifolds:

The full subcategory embedding into the category of diffeological spaces:

• Richard Hain, A characterization of smooth functions defined on a Banach space, Proc. Amer. Math. Soc. 77 (1979), 63-67 (web, pdf)

Last revised on September 19, 2021 at 06:25:40. See the history of this page for a list of all contributions to it.