Shifted Poisson structures on differentiable stacks

The purpose of this paper is to investigate shifted $(+1)$ Poisson structures in context of differential geometry. The relevant notion is shifted $(+1)$ Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equivalence of quasi-Poisson groupoids. Thus isomorphism classes of $(+1)$ Poisson stack correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following programs of independent interests: (1) We introduce a $mathbb Z$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under Morita equivalence of Lie groupoids, thus can be considered as polyvector fields on the corresponding differentiable stack ${mathfrak X}$. It turns out that shifted $(+1)$ Poisson structures on ${mathfrak X}$ correspond exactly to elements of the Maurer-Cartan moduli set of the corresponding dgla. (2) We introduce the notion of tangent complex $T_{mathfrak X}$ and cotangent complex $L_{mathfrak X}$ of a differentiable stack ${mathfrak X}$ in terms of any Lie groupoid $Gamma{rightrightarrows} M$ representing ${mathfrak X}$. They correspond to homotopy class of 2-term homotopy $Gamma$-modules $A[1]rightarrow TM$ and $T^vee Mrightarrow A^vee[-1]$, respectively. We prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${mathfrak X}$, defines a morphism ${L_{{mathfrak X}}}[1]to {T_{{mathfrak X}}}$.