# nLab concrete smooth infinity-groupoid

Contents

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

### Context

#### 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

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

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${}_{smooth}$.

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:

### 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 $f$.

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

A writeup of some aspects is in section 3.3.1 of