nLab smooth infinity-groupoid

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 mathematical folklore. A detailed proof is given 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.

Proposition

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${}_{smooth}$:

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

Remark

(Brown-Gersten property)
The homotopy descent of simplicial presheaves on the above site may conveniently be checked by a Brown-Gersten property, see there.

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

• smooth infinity-groupoid – structures

Related concepts

• cohesive (∞,1)-topos

  • discrete ∞-groupoid

  • D-topological ∞-groupoid

  • smooth $\infty$-groupoid

    • smooth 2-group

    • Kan-fibrant simplicial manifold

    • smooth spectrum

  • singular-smooth ∞-groupoid

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$\phantom{A}$

• complex analytic ∞-groupoid

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

• geometry of physics.

References

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

• Urs Schreiber, differential cohomology in a cohesive topos

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

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes – An $\infty$-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)

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

• David Carchedi, On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces (arXiv:1504.02394)

On smooth ∞-groupoids (called “differentiable stacks” there) and the smooth Oka principle:

• Adrian Clough, A Convenient Category for Geometric Topology, PhD thesis, UT Austin (2021) [pdf, pdf]

• Adrian Clough, The homotopy theory of differentiable sheaves [arXiv:2309.01757]

• Adrian Clough, The Homotopy Theory of Differentiable Sheaves, talk at Workshop on Homotopy Theory and Applications, CQTS (Oct 2023) [video:YT]