nLab concrete smooth infinity-groupoid

Redirected from "concrete smooth ∞-groupoid".

Contents

This is a sub-entry of smooth ∞-groupoid – structures. See there for more context.

Context

Cohesive $\infty$ -Toposes

Differential geometry

$\infty$ -Lie theory

Idea
Definition
Special cases

Diffeological spaces
Diffeological groupoids

Transgression of differential cocycles to mapping spaces
Related concepts
References

Idea

In every cohesive (∞,1)-topos there is a notion of concrete cohesive ∞-groupoids. Here we discuss concrete smooth ∞-groupoids. These are the higher generalization of diffeological spaces.

Definition

The general abstract definition is:

Definition

A concrete smooth $\infty$ -groupoid is a concrete cohesive ∞-groupoids in Smooth∞Grpd.

We will compare this intrinsic definition with more concrete models:

Definition

A diffeological space is a concrete sheaf on the site CartSp ${}_{smooth}$ .

Write

DiffeolSpace \hookrightarrow Sh(CartSp)

for the full subcategory on diffeological spaces. Notice that is a quasitopos.

Definition

A diffeological groupoid is an internal groupoid in the category of diffeological spaces.

(…)

Special cases

We discuss the special cases of n-truncated concrete smooth $\infty$ -groupoids:

0-truncated: diffeological space
1-truncated: diffeological groupoids.

Diffeological spaces

Proposition

Concrete smooth 0-groupoids are equivalently diffeological spaces.

More in detail, write $Conc(\tau_{\leq 0} Smooth \infty Grpd)$ for the full subcategory on the concrete 0-truncated objects. This is equivalent to the category of diffeological spaces

DiffeolSp \simeq Conc(\tau_{\leq 0} Smooth \infty Grpd) \,.

Proof

Let $X \in Sh(CartSp) \hookrightarrow Smooth \infty Grpd$ be a sheaf on $CartSp$ . The condition for it to be concrete is that the unit

X \to coDisc \Gamma X

is a monomorphism. Since monomorphisms of sheaves are detected objectwise (see category of sheaves) this is equivalent to the statement that for all $U \in CartSp$ the morphism

X(U) \simeq Smooth\infty Grpd(U, X) \to Smooth \infty Grpd(U, coDisc \Gamma X) \simeq \infty Grpd(\Gamma U, \Gamma X)

is a monomorphism of sets, where in the first step we used the (∞,1)-Yoneda lemma and in the last one the $(\Gamma \dashv coDisc)$ -adjunction.

That this morphism is indeed $\Gamma : Sh(U,X) \to Set(\Gamma(U), \Gamma(X)) \hookrightarrow \infty Grpd(\Gamma(U), \Gamma(X))$ follows by chasing the identity on $\Gamma X$ through the adjunction naturality square for any morphism $f : U \to X$

\array{ Set(\Gamma X, \Gamma X) &\stackrel{\simeq}{\to}& Sh(X, coDisc \Gamma X) \\ \downarrow^{\mathrlap{\Gamma(f)^*}} && \downarrow^{\mathrlap{f^*}} \\ Set(\Gamma U, \Gamma X) &\stackrel{\simeq}{\leftarrow}& Sh(U, coDisc \Gamma X) } \,.

So this is indeed the defining condition for concrete sheaves that defines diffeological spaces.

Corollary

The canonical embedding $SmoothMfd \hookrightarrow Smooth \infty Grpd$ from above factors through diffeological spaces: we have a sequence of full and faithful (∞,1)-functors

SmoothMfd \hookrightarrow DiffeolSp \hookrightarrow Smooth \infty Grpd \,.

Diffeological groupoids

We want to say the following

Proposition

Concrete 1-truncated smooth $\infty$ -groupoids are equivalent to diffeological groupoids, def. .

(…)

Proof

Let $A$ be 1-truncated and concrete. Then by definition there is a concrete 0-truncated object $A_0$ and an effective epimorphism $A_0 \to A$ – an atlas – , such that the (∞,1)-pullback

A_1 := A_0 \times_A A_0

is itself concrete.

Since $A$ is assumed to be 1-truncated, it follows that $A_1$ is 0-truncated. By Giraud's axioms in the (∞,1)-topos Smooth∞Grpd we have that $A$ is equivalent to the groupoid object in an (∞,1)-category $A_1 \stackrel{\to}{\to} A_0$ :

A \simeq \lim_\to (A_1 \stackrel{\to}{\to} A_0) \,.

Now by prop. both $A_0$ and $A_1$ are diffeological spaces. Hence the above exhibits $A$ as equivalent to a diffeological groupoid.

(…)

Transgression of differential cocycles to mapping spaces

For $n \in \mathbb{N}$ , write $\mathbf{B}^n U(1)_{conn} \in Smooth\infty Grpd$ for the smooth $\infty$ -groupoid given under the Dold-Kan correspondence by the Deligne complex. Over smooth manifolds this is the coefficient object for circle n-bundles with connection.

At ∞-Chern-Simons theory the following fact is proven:

Proposition

Let $\Sigma$ be a closed smooth manifold of dimension $dim \Sigma \leq n$ . Then there is an equivalence

hol : \tau_{n - dim \Sigma} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \to B^{n-dim \Sigma} U(1)

of discrete ∞-groupoids such that for $dim \sigma = n$ this computes the $n$ -volume holonomy of circle $n$ -bundles with connection.

Using concretization we want to refine this from discrete to smooth $\infty$ -groupoids.

Write $[\Sigma, \mathbf{B}^n U(1)_{conn}]$ for the internal hom in the (∞,1)-topos (see there).

We first look at this for $n = dim \Sigma$

Proposition

For $dim \Sigma = n$ there is an equivalence

hol : Conc \tau_0 [\Sigma, \mathbf{B}^nU(1)_{conn}] \to U(1)

in Smooth∞Grpd.

Proof

Since $\tau_0 [\Sigma, \mathbf{B}^n U(1)_{conn}]$ is 0-truncated, hence a sheaf, concretification is that discussed at concrete sheaves:

Conc(F)(U) = image( F(U) = \mathbf{H}(U,F) \to \mathbf{H}(U, coDisc \Gamma F) = Set(\Gamma(U), \Gamma(F)) ) \,.

So for $U = *$ first of all we have by prop. that

Conc \tau_0 \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})(*) = U(1)

in $Set$ .

Generally $Conc \tau_0 \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})(U)$ is therefore a subset of the set of functions of sets $U \to U(1)$ . We need to show that it is precisely the set of smooth such functions.

But this is clear: holonomy of a smoth family of smoth circle $n$ -bundles is a smooth function. Moreover, every smooth function arises this way: for $f : U \to U(1)$ any smooth function, pick a trivial family of trivial circle $n$ -bundles with connection and then rescale the connection form using $f$ .

Related concepts

References

Many of the ideas involved here are due to Dave Carchedi.

A writeup of some aspects is in section 3.3.1 of

Urs Schreiber, differential cohomology in a cohesive topos

Last revised on June 29, 2011 at 12:13:50. See the history of this page for a list of all contributions to it.

nLab concrete smooth infinity-groupoid

Context

Cohesive ∞\infty-Toposes

Differential geometry

∞\infty-Lie theory

Contents

Idea

Definition

Definition

Definition

Definition

Special cases

Diffeological spaces

Proposition

Proof

Corollary

Diffeological groupoids

Proposition

Proof

Transgression of differential cocycles to mapping spaces

Proposition

Proposition

Proof

Related concepts

References

Cohesive $\infty$ -Toposes

$\infty$ -Lie theory