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smooth infinity-groupoid

Context

Cohesive $∞$ -Toposes

Differential geometry

$∞$ -Lie theory

Idea
Definition
Properties
Structures in the cohesive $(∞, 1)$ -topos $Smooth ∞ Grpd$
Related concepts
References

Idea

A smooth $∞$ -groupoid is an ∞-groupoid equipped with cohesion in the form of smooth structure. Examples include smooth manifolds, Lie groups and Lie groupoids.

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

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

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

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 $∐_{i} U_{i}$ is paracompact. Hence we may find a differentiably good open cover ${K_{j} \to ∐_{i} U_{i}}$ . 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 $∞$ -groupoids is the (∞,1)-category of (∞,1)-sheaves on CartSp $_{smooth}$ :

Smooth ∞ Grpd : = {Sh}_{(∞, 1)} ({CartSp}_{smooth}) .

Properties

Cohesion

Proposition

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

Proof

The site CartSp $_{smooth}$ is (as discussed there) an ∞-cohesive site (see there).

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 $Smooth ∞ Grpd$ is equivalent to the hypercompletion ${\hat{Sh}}_{(∞, 1)} (SmoothMfg)$ of the (∞,1)-category of (∞,1)-sheaves on the large site SmoothMfd

Smooth ∞ Grpd ≃ {\hat{Sh}}_{(∞, 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 $Smooth ∞ Grpd$ is a full and faithful (∞,1)-functor

SmoothMfd ↪ Smooth ∞ Grpd, .

Relative cohesion

We discuss the relation of $Smooth ∞ Grpd$ 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_{!} ⊣ i^{*} ⊣ i_{*}) : Smooth ∞ Grpd \overset{\overset{i_{!}}{\to}}{\overset{\overset{i^{*}}{\leftarrow}}{\underset{i_{*}}{\to}}} ETop ∞ Grpd

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

\begin{matrix} SmoothMfd & ↪ & Smooth ∞ Grpd \\ ↓^{i} & ↓^{i_{!}} \\ Mfd & ↪ & ETop ∞ Grpd \end{matrix}

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 $Smooth ∞ Grpd$ factors through that of ETop∞Grpd

(Π_{Smooth} ⊣ {Disc}_{Smooth} ⊣ Γ_{Smooth}) : Smooth ∞ Grpd \overset{\overset{i_{!}}{\to}}{\overset{\overset{i^{*}}{\leftarrow}}{\underset{i_{*}}{\to}}} ETop ∞ Grpd \overset{\overset{Π_{ETop}}{\to}}{\overset{\overset{{Disc}_{ETop}}{\leftarrow}}{\underset{Γ_{ETop}}{\to}}} ∞ 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 ( $C^{∞}$ -rings). Write $SmoothAlg : = Alg (C)$ for the category of its algebras. Let $InfPoint ↪ {SmoothAlg}^{op}$ be the full subcategory on the infinitesimally thickened points.

Definition

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

i : {CartSp}_{smooth} ↪ {CartSp}_{synthdiff}

for the canonical inclusion.

Proposition

The inclusion exhibits an infinitesimal cohesive neighbourhood of $Smooth ∞ Grpd$

(i_{!} ⊣ i^{*} ⊣ i_{*} ⊣ i^{!}) : Smooth ∞ Grpd ↪ SynthDiff ∞ 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 $∞$ -groupoids is

Sh (CartSp) ≃ Smooth ∞ {Grpd}_{\leq 0} ↪ Smooth ∞ Grpd,

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

The concrete objects in there

Smooth ∞ {Grpd}_{\leq 0}^{conc} ↪ Smooth ∞ Grpd

are precisely the diffeological spaces.

Structures in the cohesive $(∞, 1)$ -topos $Smooth ∞ Grpd$

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

This section is at

smooth infinity-groupoid -- structures

Related concepts

cohesive (∞,1)-topos
- discrete ∞-groupoid
- continuous ∞-groupoid
- smooth $∞$ -groupoid
- synthetic differential ∞-groupoid

References

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

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

Urs Schreiber, differential cohomology in a cohesive topos

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

Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes – An $∞$ -Lie theoretic construction (web).

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

Jean-Luc Brylinski, Differentiable Cohomology of Gauge Groups (arXiv)

which parallels

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

A review is in section 4 of

Chris Schommer-Pries, A finite-dimensional String 2-group (arXiv:0911.2483)

Classification of topological principal 2-bundles is discussed in

John Baez, Danny Stevenson, The classifying space of a topological 2-group Algebraic Topology Abel Symposia, 2009, Volume 4, 1-31 (arXiv:0801.3843)

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

Thomas Nikolaus, Konrad Waldorf, Four Equivalent Versions of Non-Abelian Gerbes (arXiv:1103.4815)

Revised on February 27, 2012 20:12:31 by David Corfield (129.12.18.29)

nLab smooth infinity-groupoid

Context

Cohesive ∞-Toposes

Backround

Definition

Presentation over a site

Structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

Models

Differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

∞-Lie groupoids

∞-Lie groups

∞-Lie algebroids

∞-Lie algebras

Contents

Idea

Definition

Definition

Proposition

Proof

Definition

Observation

Proof

Definition

Definition

Properties

Cohesion

Proposition

Proof

Definition

Proposition

Proof

Proposition

Proof

Corollary

Relative cohesion

Continuous cohesion

Proposition

Proof

Corollary

Proof

Observation

Infinitesimal cohesion

Definition

Proposition

Proof

Truncations

Structures in the cohesive (∞,1)-topos Smooth∞Grpd

Related concepts

References

nLab
smooth infinity-groupoid

Cohesive $∞$ -Toposes

Structures in a cohesive $(∞, 1)$ -topos

$∞$ -Lie theory

$∞$ -Lie groupoids

$∞$ -Lie groups

$∞$ -Lie algebroids

$∞$ -Lie algebras

Structures in the cohesive $(∞, 1)$ -topos $Smooth ∞ Grpd$