# nLab SmoothManifolds

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)

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Definition

###### Definition

$SmoothManifolds$ is the category whose

Similarly, for $n \in \mathbb{N}$

$DifferentiableManifolds_n$ is the category whose

for $n$-fold differentiabiliy.

Each of these categories is also commonly denoted $Man$ or $Mfd$ or Diff etc.

## Properties

### As a site

###### Proposition

The category $SmoothManifolds$ becomes a large site by equipping it with the coverage consisting of open covers.

This is an essentially small site: a dense sub-site for $SmoothManifolds$ is given by CartSp${}_{smooth}$.

###### Proof

The first statement follows trivially as for Top: the preimage of an open subset under a continuous function is again open (by definition of continuouss function).

For the second statement one needs that every paracompact manifold admits a differentially good open cover : an open cover by open balls that are diffeomorphic to a Cartesian spaces. The proof for this is spelled out at good open cover.

###### Corollary

The sheaf topos over $SmoothManifolds$ is a cohesive topos.

The hypercompletion of the (∞,1)-sheaf (∞,1)-topos over $SmoothManifolds$ is a cohesive (∞,1)-topos.

###### Proof

For the first statement, use that by the comparison lemma discussed at dense sub-site we have an equivalence of categories

$Sh(SmoothManifolds) \simeq Sh(CartSp_{smooth}) \,.$

By the discussion at CartSp we have that $CartSp_{smooth}$ is a cohesive site. By the discussion there the claim follows.

For the second statement observe that the Joyal-Jardine model structure on simplicial sheaves $Sh(SmoothManifolds)^{\Delta^{op}}_{loc}$ is a presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves $\hat Sh_{(\infty,1)}(SmoothManifolds)$ (see presentations of (∞,1)-sheaf (∞,1)-toposes). By the above result it follows that there is an equivalence of (∞,1)-categories between the hypercompletions

$\hat Sh_{(\infty,1)}(SmoothManifolds) \simeq \hat Sh_{(\infty,1)}(CartSp_{smooth}) \,.$

Now CartSp${}_{smooth}$ is even an ∞-cohesive site. By the discussion there it follows that $Sh_{(\infty,1)}(CartSp_{smooth})$ (before hypercompletion) is a cohesive (∞,1)-topos. This means that it is in particular a local (∞,1)-topos. But this implies (as discussed there), that the (∞,1)-category of (∞,1)-sheaves already is the hypercomplete (∞,1)-topos. Therefore finally

$\cdots \simeq Sh_{(\infty,1)}(CartSp_{smooth}) \,.$
###### Remark

The cohesive topos $Sh(SmoothManifolds) \simeq Sh(CartSp_{smooth})$ is in particular the home of diffeological spaces. See there for more details.

$Smooth \infty Grp := Sh_{(\infty,1)}(SmoothManifolds) \simeq Sh_{(\infty,1)}(CartSp_{smooth})$

is that of smooth ∞-groupoids. Discussed at Smooth∞Grpd.

The theory of differentiable stacks is that of geometric stacks in the (2,1)-sheaf (2,1)-topos

$Sh_{(2,1)}(SmoothManifolds) \simeq Sh_{(2,1)}(CartSp_{smooth}) \simeq \tau_{\leq 1} Sh_{(\infty,1)}(CartSp_{smooth})$
• CartSp${}_{top}$ , TopMfd

• CartSp${}_{smth}$

• Diff

category: category

Last revised on June 26, 2022 at 11:54:34. See the history of this page for a list of all contributions to it.