# nLab infinite-dimensional 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

The basic definition of a manifold (especially a smooth manifold) is as a space locally modeled on a finite-dimensional Cartesian space. This can be generalized to a notion of smooth manifolds locally modeled on infinite-dimensional topological vector spaces. Typical examples of these are mapping spaces between finite-dimensional manifolds, such as loop spaces.

## Definitions

See specific versions:

## Properties: Embedding into convenient toposes

Various types of smooth manifolds embed into the quasi-toposes of diffeological spaces and hence the topos of smooth spaces. See there for more.

### General

Modern textbook account in terms of Frölicher spaces and diffeological spaces:

Aspects of the homotopy theory of infinite-dimensional manifolds:

### Integration

On integration over infinite-dimensional manifolds (for instance path integrals):

• Irving Segal, Algebraic integration theory, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489 (Euclid)

• Hui-Hsiung Kuo, Integration theory on infinite-dimensional manifolds, Transactions of the American Mathematical Society Vol. 159, (Sep., 1971), pp. 57-78 (JSTOR)

• David Shale, Invariant integration over the infinite dimensional orthogonal group and related spaces, Transactions of the American Mathematical Society Vol. 124, No. 1 (Jul., 1966), pp. 148-157 (JSTOR)