A Chern-Weil theory for Lie groupoids with Cartan connections

We extend Chern-Weil theory to Lie groupoids carrying a Cartan connection. For a principal $(Σ, ω)$-bundle $(P, θ)$ over $X$ we construct a map of differential graded algebras from an $ω$-basic subcomplex of the Weil algebra of the Lie algebroid of $Σ$ to $Ω^\bullet(X)$, whose map in cohomology does