nLab concrete smooth infinity-groupoid

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

---

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

• concrete sheaf, concrete (∞,1)-sheaf

• diffeological space, Lie groupoid

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