Transitive double Lie algebroids via core diagrams

<p style='text-indent:20px;'>The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called <i>transitive</i>. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.</p><p style='text-indent:20px;'>The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.</p><p style='text-indent:20px;'>Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.</p>

On Rham cohomology of locally trivial Lie groupoids over triangulated manifolds

Based on the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology of a transitive Lie algebroid, it is proved that the Rham cohomology of a locally trivial Lie groupoid G on a smooth manifold M is isomorphic to the piecewise Rham cohomology of G, in which G and M are manifolds without boundary and M is smoothly triangulated by a finite simplicial complex K such that, for each simplex ∆ of K, the inverse images of ∆ by the source and target mappings of G are transverses submanifolds in the ambient space G. As a consequence, it is shown that the piecewise de Rham cohomology of G does not depend on the triangulation of the base.

Shifted Poisson Structures on Differentiable Stacks

The purpose of this paper is to investigate $(+1)$-shifted Poisson structures in the context of differential geometry. The relevant notion is that of $(+1)$-shifted Poisson structures on differentiable stacks. More precisely, we develop the notion of the Morita equivalence of quasi-Poisson groupoids. Thus, isomorphism classes of $(+1)$-shifted Poisson stacks correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following program, which is of independent interest: (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 the Morita equivalence of Lie groupoids, and thus they can be considered to be polyvector fields on the corresponding differentiable stack ${\mathfrak{X}}$. It turns out that $(+1)$-shifted 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 the tangent complex $T_{\mathfrak{X}}$ and the 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 a homotopy class of 2-term homotopy $\Gamma$-modules $A[1]\rightarrow TM$ and $T^{\vee } M\rightarrow A^{\vee }[-1]$, respectively. Relying on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids, we prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${\mathfrak{X}}$ defines a morphism ${L_{\mathfrak{X}}}[1]\to{T_{\mathfrak{X}}}$.