# nLab smooth infinity-groupoid

Contents

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $(\infty,1)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion?

## Models

#### 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

## Idea

A smooth $\infty$-groupoid is an ∞-groupoid equipped with cohesion in the form of smooth structure. Examples include smooth manifolds, Lie groups, Lie groupoids and generally Lie infinity-groupoids, but also for instance moduli spaces of differential forms, moduli stacks of principal connections and generally of cocycles in differential cohomology.

The (∞,1)-topos $Smooth \infty Grpd$ of all smooth $\infty$-groupoids is a cohesive (∞,1)-topos. It realizes a higher geometry version of differential geometry.

Many properties of smooth $\infty$-groupoids are inherited from the underlying Euclidean-topological ∞-groupoids. See ETop∞Grpd for more.

There is a refinement of smooth $\infty$-groupoids to synthetic differential ∞-groupoids. See SynthDiff∞Grpd for more on that.

## Definition

###### Definition

For $X$ a smooth manifold, say an open cover $\{U_i \to X\}$ is a differentiably good open cover if each non-empty finite intersection of the $U_i$ is diffeomorphic to a Cartesian space.

###### Proposition

Every paracompact smooth manifold admits a differentiably good open cover.

###### Proof

This is a folk theorem. A detailed proof is at good open cover.

###### Definition

Let SmoothMfd be the large site of paracompact smooth manifolds with smooth functions between them and equipped with the coverage of differentiably good open covers.

###### Observation

This does indeed define a coverage. The Grothendieck topology that is generated from it is the standard open cover topology.

###### Proof

For $\{U_i \to X\}$ any open cover of a paracompact manifold also $\coprod_i U_i$ is paracompact. Hence we may find a differentiably good open cover $\{K_j \to \coprod_i U_i\}$. This is then a refinement of the original open cover of $X$.

###### Definition

Let CartSp${}_{smooth}$ be the site of Cartesian spaces with smooth functions between them and equipped with the coverage of differentiably good open covers.

###### Definition

The (∞,1)-topos of smooth $\infty$-groupoids is the (∞,1)-category of (∞,1)-sheaves on CartSp${}_{smooth}$:

$Smooth \infty Grpd := Sh_{(\infty,1)}(CartSp_{smooth}) \,.$

## Properties

### Cohesion

###### Proposition

$Smooth \infty Grpd$ is a cohesive (∞,1)-topos.

###### Proof

The site CartSp${}_{smooth}$ is (as discussed there) an ∞-cohesive site. See there for the implication.

###### Definition

Let SmoothMfd be the large site of paracompact smooth manifolds with smooth functions between them and equipped with the coverage whose covering families are differentiably good open covers : open covers $\{U_i \to U\}$ where each non-empty open intersection is diffeomorphic to a Cartesian space.

###### Proposition

This does indeed define a coverage and the Grothendieck topology generated by it is the standard open cover topology.

###### Proof

This is discussed in detail at good open cover.

###### Proposition

The (∞,1)-topos $Smooth \infty Grpd$ is equivalent to the hypercompletion $\hat Sh_{(\infty,1)}(SmoothMfd)$ of the (∞,1)-category of (∞,1)-sheaves on the large site SmoothMfd

$Smooth \infty Grpd \simeq \hat Sh_{(\infty,1)}(SmoothMfd) \,.$
###### Proof

By the above we have that CartSp${}_{smooth}$ is a dense sub-site of SmoothMfd. With this the claim follows as in the analogous discussion at ETop∞Grpd.

###### Corollary

The canonical embedding of smooth manifolds as 0-truncated objects in $Smooth\infty Grpd$ is a full and faithful (∞,1)-functor

$SmoothMfd \hookrightarrow Smooth \infty Grpd ,.$

### Relative cohesion

We discuss the relation of $Smooth\infty Grpd$ to other cohesive (∞,1)-toposes.

#### Continuous cohesion

The cohesive (∞,1)-topos ETop∞Grpd of Euclidean-topological ∞-groupoids has as site of definition CartSp${}_{top}$. There is a canonical forgetful functor

$i : CartSp_{smooth} \to CartSp_{top}$
###### Proposition

The functor $i$ extends to an essential (∞,1)-geometric morphism

$(i_! \dashv i^* \dashv i_*) : Smooth\infty Grpd \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}} ETop\infty Grpd$

such that the (∞,1)-Yoneda embedding is factored through the induced inclusion SmoothMfd $\stackrel{i}{\hookrightarrow}$ Mfd as

$\array{ SmoothMfd &\hookrightarrow& Smooth\infty Grpd \\ \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{i_!}} \\ Mfd &\hookrightarrow& ETop\infty Grpd } \,$
###### Proof

Using the observation that $i$ preserves coverings and pullbacks along morphism in covering families, the proof follows precisely the steps of the proof of this proposition.

(Both of these are special cases of a general statement about morphisms of (∞,1)-sites, which should eventually be stated in full generality somewhere).

###### Corollary

The essential global section (∞,1)-geometric morphism of $Smooth \infty Grpd$ factors through that of ETop∞Grpd

$(\Pi_{Smooth} \dashv Disc_{Smooth} \dashv \Gamma_{Smooth}) : Smooth \infty Grpd \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}} ETop\infty Grpd \stackrel{\overset{\Pi_{ETop}}{\to}}{\stackrel{\overset{Disc_{ETop}}{\leftarrow}}{\underset{\Gamma_{ETop}}{\to}}} \infty Grpd$
###### Proof

This follows from the essential uniqueness of the global section (∞,1)-geometric morphism and of adjoint (∞,1)-functors.

###### Observation

The functor $i_!$ here is the forgetful functor that forgets smooth structure and only remembers Euclidean topology-structure.

#### Infinitesimal cohesion

Observe that CartSp${}_{smooth}$ is (the syntactic category of) a Lawvere theory: the algebraic theory of smooth algebras ($C^\infty$-rings). Write $SmoothAlg := Alg(C)$ for the category of its algebras. Let $InfPoint \hookrightarrow SmoothAlg^{op}$ be the full subcategory on the infinitesimally thickened points.

###### Definition

Let CartSp${}_{synthdiff} \hookrightarrow SmoothAlg^{op}$ be the full subcategory on the objects of the form $U \times D$ with $D \in CartSp_{smooth} \hookrightarrow SmoothAlg^{op}$ and $D \in InfPoint \hookrightarrow SmoothAlg^{op}$. Write

$i : CartSp_{smooth} \hookrightarrow CartSp_{synthdiff}$

for the canonical inclusion.

###### Proposition

The inclusion exhibits an infinitesimal cohesive neighbourhood of $Smooth \infty Grpd$

$(i_! \dashv i^* \dashv i_* \dashv i^!) : Smooth \infty Grpd \hookrightarrow SynthDiff\infty Grpd \,,$

where SynthDiff∞Grpd is the cohesive (∞,1)-topos of synthetic differential ∞-groupoids: the (∞,1)-category of (∞,1)-sheaves over $CartSp_{synthdiff}$.

###### Proof

This follows as a special case of this proposition after observing that $CartSp_{synthdiff}$ is an infinitesimal neighbourhood site of $CartSp_{smooth}$ in the sense defined there.

In SynthDiff∞Grpd we have ∞-Lie algebras and ∞-Lie algebroids as actual infinitesimal objects. See there for more details.

### Truncations

The (1,1)-topos on the 0-truncated smooth $\infty$-groupoids is

$Sh(CartSp) \simeq Smooth \infty Grpd_{\leq 0} \hookrightarrow Smooth\infty Grpd \,,$

the sheaf topos on SmthMfd/CartSp discussed at smooth space.

The concrete objects in there

$Smooth\infty Grpd_{\leq 0}^{conc} \hookrightarrow Smooth \infty Grpd$

are precisely the diffeological spaces.

## Structures in the cohesive $(\infty,1)$-topos $Smooth \infty Grpd$

We discuss the general abstract structures in a cohesive (∞,1)-topos realized in $Smooth \infty Grpd$.

This section is at

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$$\phantom{A}$site$\phantom{A}$$\phantom{A}$sheaf topos$\phantom{A}$$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$$\phantom{A}$Point$\phantom{A}$$\phantom{A}$Set$\phantom{A}$$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$CartSp$\phantom{A}$$\phantom{A}$SmoothSet$\phantom{A}$$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$$\phantom{A}$FormalCartSp$\phantom{A}$$\phantom{A}$FormalSmoothSet$\phantom{A}$$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$SuperFormalCartSp$\phantom{A}$$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

Smooth $\infty$-groupoids and related cohesive structures play a central role in the discussion at

For standard references on differential geometry and Lie groupoids see there.

The $(\infty,1)$-topos $Smooth \infty Grpd$ is discussed in section 3.3 of

A discussion of smooth $\infty$-groupoids as $(\infty,1)$-sheaves on $CartSp$ and the presentaton of the $\infty$-Chern-Weil homomorphism on these is in

For references on Chern-Weil theory in Smooth∞Grpd and connection on a smooth principal ∞-bundle, see there.

The results on differentiable Lie group cohomology used above are in

• P. Blanc, Cohomologie différentiable et changement de groupes Astérisque, vol. 124-125 (1985), pp. 113-130.

and

which parallels

• Graeme Segal, Cohomology of topological groups , Symposia Mathematica, Vol IV (1970) (1986?) p. 377

A review is in section 4 of

Classification of topological principal 2-bundles is discussed in

and the generalization to classification of smooth principal 2-bundles is in

Further discussion of the shape modality on smooth $\infty$-groupoids is in