This is a sub-entry of smooth ∞-groupoid -- structures. See there for more context.
Formal Lie groupoids
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:
We will compare this intrinsic definition with more concrete models:
We discuss the special cases of n-truncated concrete smooth -groupoids:
Let be a sheaf on . 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 the morphism
is a monomorphism of sets, where in the first step we used the (∞,1)-Yoneda lemma and in the last one the -adjunction.
That this morphism is indeed follows by chasing the identity on through the adjunction naturality square for any morphism
So this is indeed the defining condition for concrete sheaves that defines diffeological spaces.
The canonical embedding 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 -groupoids are equivalent to diffeological groupoids, def. 3.
Transgression of differential cocycles to mapping spaces
For , write for the smooth -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 be a closed smooth manifold of dimension . Then there is an equivalence
of discrete ∞-groupoids such that for this computes the -volume holonomy of circle -bundles with connection.
Using concretization we want to refine this from discrete to smooth -groupoids.
Write for the internal hom in the (∞,1)-topos (see there).
We first look at this for
For there is an equivalence
Since is 0-truncated, hence a sheaf, concretification is that discussed at concrete sheaves:
So for first of all we have by prop. 3 that
Generally is therefore a subset of the set of functions of sets . 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 -bundles is a smooth function. Moreover, every smooth function arises this way: for any smooth function, pick a trivial family of trivial circle -bundles with connection and then rescale the connection form using .
Many of the ideas involved here are due to Dave Carchedi.
A writeup of some aspects is in section 3.3.1 of