Let $X$ be a differentiable stacks, and let $(G_{0}, G_{1}, s,t)$ be a Lie groupoid representing $X$. Let $NG_{\bullet}$ be the nerve of the above groupoid. The De rham complex of $X$ can be defined as the total complex of the cosimplicial dg algebra $A_{DR}^{*}(NG_{\bullet})$ (see https://www.math.ubc.ca/~behrend/CohSta-1.pdf), where $A_{DR}(-)$ is the ordinary De Rham functor on smooth manifolds. I wonder if there is a notion of integration along the fiber (of representable morphisms)/pusforward in these settings. More concretely: for a smooth fiber bundle map $p\: : \: E\to M$ with dimension $k$ (assuming that the fiber has no boundary) we may define a map $\int_{p}\: : \: A_{DR}(E)\to A_{DR}(M)$ of degree $k$ given by "vertical" integration. This map induces the push forward $p_{*}$ in cohomology. Is there a way to define the integration along the fiber between Lie groupoids? References?
P.S: A definition of vector bundle on topological stack may be found at http://arxiv.org/abs/0712.3857.
