# nLab convenient manifold

### 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

graded differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

A manifold, possibly infinite-dimensional, is called a convenient manifold – implicitly meaning: convenient for differential geometry – if it is modeled on a convenient vector space.

One should note that this usage of the adjective ‘convenient’ is different to that in ‘convenient category’, for example of smooth spaces. In that case the category is convenient, whereas here the objects are convenient.

## Properties

### Embedding into the Cahiers-topos

Together with convenient vector spaces, convenient manifods embed into the Cahier topos of synthetic differential smooth spaces. See at Cahiers topos for more on this.

## References

A standard textbook reference is

• Peter Michor, A convenient setting for differential geometry and global analysis, Cahiers Topologie Géom. Différentielle 25 (1984), no. 1, 63–109, MR86g:58014a; A convenient setting for differential geometry and global analysis. II, Cahiers Topologie Géom. Différentielle 25 (1984), no. 2, 113–178.

A survey is for instance in the slides

• Richard Blute, Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic (2011) (pdf)
• John C. Baez, Alexander E. Hoffnung, Convenient categories of smooth spaces, Trans. Amer. Math. Soc. 363 (2011), 5789-5825 pdf