This is a sub-entry of smooth ∞-groupoid -- structures. See there for more context.
structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
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.
The general abstract definition is:
A concrete smooth $\infty$-groupoid is a concrete cohesive ∞-groupoids in Smooth∞Grpd.
We will compare this intrinsic definition with more concrete models:
A diffeological space is a concrete sheaf on the site CartSp${}_{smooth}$.
Write
for the full subcategory on diffeological spaces. Notice that is a quasitopos.
A diffeological groupoid is an internal groupoid in the category of diffeological spaces.
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We discuss the special cases of n-truncated concrete smooth $\infty$-groupoids:
0-truncated: diffeological space
1-truncated: diffeological groupoids.
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
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
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
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 $f : U \to X$
So this is indeed the defining condition for concrete sheaves that defines diffeological spaces.
The canonical embedding $SmoothMfd \hookrightarrow Smooth \infty Grpd$ from above factors through diffeological spaces: we have a sequence of full and faithful (∞,1)-functors
We want to say the following
Concrete 1-truncated smooth $\infty$-groupoids are equivalent to diffeological groupoids, def. 3.
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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
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 $A_1 \stackrel{\to}{\to} A_0$:
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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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:
Let $\Sigma$ be a closed smooth manifold of dimension $dim \Sigma \leq n$. Then there is an equivalence
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 $n = dim \Sigma$
For $dim \Sigma = n$ there is an equivalence
in Smooth∞Grpd.
Since $\tau_0 [\Sigma, \mathbf{B}^n U(1)_{conn}]$ is 0-truncated, hence a sheaf, concretification is that discussed at concrete sheaves:
So for $U = *$ first of all we have by prop. 3 that
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