Schreiber Chern-Weil theory in a smooth (∞,1)-topos

under construction

Ordinary Chern-Weil theory provides for $G$ a Lie group and for any connection $\nabla$ on a smooth $G$-principal bundle $P \to X$ a morphism of dg-algebras

$$\Omega^\bullet(X) \leftarrow inv(\mathfrak{g}) : P(\nabla) \,,$$

where $inv(\mathfrak{g})$ is the dg-algebra of invariant polynomials on the Lie algebra $\mathfrak{g}$ of $G$: the image of a given invariant polynomial is the corresponding curvature? characteristic class of the bundle.

More generally, in a smooth (∞,1)-topos we have the notion of Ehresmann ∞-connection on a $A$-principal ∞-bundle. For $X$ 0-truncated this comes with a morphism

$$\Omega^\bullet(X) \leftarrow inv(\mathfrak{a}) : P(\nabla)$$

from the invariant polynomial on the ∞-Lie algebroid $\mathfrak{a}$ of $A$.