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concrete smooth infinity-groupoid

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


Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

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

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

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{}_{smooth}.

Write

DiffeolSpaceSh(CartSp) 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:

Diffeological spaces

Proposition

Concrete smooth 0-groupoids are equivalently diffeological spaces.

More in detail, write Conc(τ 0SmoothGrpd)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

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

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

XcoDiscΓX 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 UCartSpU \in CartSp the morphism

X(U)SmoothGrpd(U,X)SmoothGrpd(U,coDiscΓX)Grpd(ΓU,ΓX) 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 (ΓcoDisc)(\Gamma \dashv coDisc)-adjunction.

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

Set(ΓX,ΓX) Sh(X,coDiscΓX) Γ(f) * f * Set(ΓU,ΓX) Sh(U,coDiscΓX). \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 SmoothMfdSmoothGrpdSmoothMfd \hookrightarrow Smooth \infty Grpd from above factors through diffeological spaces: we have a sequence of full and faithful (∞,1)-functors

SmoothMfdDiffeolSpSmoothGrpd. 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. 3.

(…)

Proof

Let AA be 1-truncated and concrete. Then by definition there is a concrete 0-truncated object A 0A_0 and an effective epimorphism A 0AA_0 \to A – an atlas – , such that the (∞,1)-pullback

A 1:=A 0× AA 0 A_1 := A_0 \times_A A_0

is itself concrete.

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

(1)Alim (A 1A 0). A \simeq \lim_\to (A_1 \stackrel{\to}{\to} A_0) \,.

Now by prop. 1 both A 0A_0 and A 1A_1 are diffeological spaces. Hence the above exhibits AA as equivalent to a diffeological groupoid.

(…)

Transgression of differential cocycles to mapping spaces

For nn \in \mathbb{N}, write B nU(1) connSmoothGrpd\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 Σ\Sigma be a closed smooth manifold of dimension dimΣndim \Sigma \leq n. Then there is an equivalence

hol:τ ndimΣH(Σ,B nU(1) conn)B ndimΣU(1) 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 dimσ=ndim \sigma = n this computes the nn-volume holonomy of circle nn-bundles with connection.

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

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

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

Proposition

For dimΣ=ndim \Sigma = n there is an equivalence

hol:Concτ 0[Σ,B nU(1) conn]U(1) hol : Conc \tau_0 [\Sigma, \mathbf{B}^nU(1)_{conn}] \to U(1)

in Smooth∞Grpd.

Proof

Since τ 0[Σ,B nU(1) conn]\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)=H(U,F)H(U,coDiscΓF)=Set(Γ(U),Γ(F))). 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=*U = * first of all we have by prop. 3 that

Concτ 0H(Σ,B nU(1) conn)(*)=U(1) Conc \tau_0 \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})(*) = U(1)

in SetSet.

Generally Concτ 0H(Σ,B nU(1) conn)(U)Conc \tau_0 \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})(U) is therefore a subset of the set of functions of sets UU(1)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 nn-bundles is a smooth function. Moreover, every smooth function arises this way: for f:UU(1)f : U \to U(1) any smooth function, pick a trivial family of trivial circle nn-bundles with connection and then rescale the connection form using ff.

References

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

A writeup of some aspects is in section 3.3.1 of

Revised on June 29, 2011 12:13:50 by Urs Schreiber (131.211.238.160)