nLab
concrete smooth infinity-groupoid

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

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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 $\mathrm{\infty }$-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${}_{\mathrm{smooth}}$.

Write

$$\mathrm{DiffeolSpace}\hookrightarrow \mathrm{Sh}(\mathrm{CartSp})$$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.

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Special cases

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

• 0-truncated: diffeological space

• 1-truncated: diffeological groupoids.

Diffeological spaces

Proposition

Concrete smooth 0-groupoids are equivalently diffeological spaces.

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

$$\mathrm{DiffeolSp}\simeq \mathrm{Conc}({\tau }_{\le 0}\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}).$$DiffeolSp \simeq Conc(\tau_{\leq 0} Smooth \infty Grpd) \,.

Proof

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

$$X\to \mathrm{coDisc}\Gamma X$$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 $U\in \mathrm{CartSp}$ the morphism

$$X(U)\simeq \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(U,X)\to \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(U,\mathrm{coDisc}\Gamma X)\simeq \mathrm{\infty }\mathrm{Grpd}(\Gamma U,\Gamma X)$$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 $(\Gamma ⊣\mathrm{coDisc})$-adjunction.

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

$$\begin{array}{ccc}\mathrm{Set}(\Gamma X,\Gamma X)& \stackrel{\simeq }{\to }& \mathrm{Sh}(X,\mathrm{coDisc}\Gamma X)\\ {\downarrow }^{\Gamma (f{)}^{*}}& & {\downarrow }^{f^{*}}\\ \mathrm{Set}(\Gamma U,\Gamma X)& \stackrel{\simeq }{\leftarrow }& \mathrm{Sh}(U,\mathrm{coDisc}\Gamma X)\end{array}.$$\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 $\mathrm{SmoothMfd}\hookrightarrow \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ from above factors through diffeological spaces: we have a sequence of full and faithful (∞,1)-functors

$$\mathrm{SmoothMfd}\hookrightarrow \mathrm{DiffeolSp}\hookrightarrow \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}.$$SmoothMfd \hookrightarrow DiffeolSp \hookrightarrow Smooth \infty Grpd \,.

Diffeological groupoids

We want to say the following

Proposition

Concrete 1-truncated smooth $\mathrm{\infty }$-groupoids are equivalent to diffeological groupoids, def. 3.

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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

$$A_1:=A_0{\times }_A{A}_0$$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 $A_1\overrightarrow{\to }{A}_0$:

(1)$$A\simeq \underrightarrow{\lim }(A_1\overrightarrow{\to }{A}_0).$$A \simeq \lim_\to (A_1 \stackrel{\to}{\to} A_0) \,.

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

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Transgression of differential cocycles to mapping spaces

For $n\in \mathbb{N}$, write $B^{n}U(1{)}_{\mathrm{conn}}\in \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ for the smooth $\mathrm{\infty }$-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 $\dim \Sigma \le n$. Then there is an equivalence

$$\mathrm{hol}:{\tau }_{n-\dim \Sigma }H(\Sigma ,B^{n}U(1{)}_{\mathrm{conn}})\to B^{n-\dim \Sigma }U(1)$$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 $\dim \sigma =n$ this computes the $n$-volume holonomy of circle $n$-bundles with connection.

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

Write $[\Sigma ,B^{n}U(1{)}_{\mathrm{conn}}]$ 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

$$\mathrm{hol}:\mathrm{Conc}{\tau }_0[\Sigma ,B^{\mathrm{nU}}(1{)}_{\mathrm{conn}}]\to U(1)$$hol : Conc \tau_0 [\Sigma, \mathbf{B}^nU(1)_{conn}] \to U(1)

in Smooth∞Grpd.

Proof

Since ${\tau }_0[\Sigma ,B^{n}U(1{)}_{\mathrm{conn}}]$ is 0-truncated, hence a sheaf, concretification is that discussed at concrete sheaves:

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

So for $U=*$ first of all we have by prop. 3 that

$$\mathrm{Conc}{\tau }_{0}H(\Sigma ,B^{n}U(1{)}_{\mathrm{conn}})(*)=U(1)$$Conc \tau_0 \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})(*) = U(1)

in $\mathrm{Set}$.

Generally $\mathrm{Conc}{\tau }_{0}H(\Sigma ,B^{n}U(1{)}_{\mathrm{conn}})(U)$ 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