Given a Lie groupoid G over a manifold M , we show that multiplicative 2forms on G relatively closed with respect to a closed 3-form φ on M correspond to maps from the Lie algebroid of G into T * M satisfying an algebraic condition and a differential condition with respect to the φ-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to φ-twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-hamiltonian spaces and group-valued momentum maps.Motivated by the relationship between symplectic realizations of Poisson manifolds and hamiltonian actions [9], we study presymplectic realizations of twisted Dirac structures. Just as in usual Poisson geometry, these presymplectic realizations carry natural actions of presymplectic groupoids. In fact, it is this property that determines our definition of presymplectic realizations. An important example of twisted Dirac structure is described in [27, Example 4.2]: any nondegenerate invariant inner product on the Lie algebra h of a Lie group H induces a natural Dirac structure on H, twisted by the invariant Cartan 3-form; we call such structures Cartan-Dirac structures. We show that presymplectic realizations of Cartan-Dirac structures are equivalent to the quasi-hamiltonian h-spaces of Alekseev, Malkin and Meinrenken [1] in such a way that the realization maps are the associated groupvalued momentum maps. It also follows from our results that the transformation groupoid H ⋉ H corresponding to the conjugation action carries a canonical twisted presymplectic structure, which we obtain explicitly by "integrating" the Cartan-Dirac structure. As a result, we recover the 2-form on the "double" D(H) of [1] and the AMM groupoid of [4]. (Closely related forms were introduced earlier in [18,30].) A unifying approach to momentum map theories based on Morita equivalence of presymplectic groupoids has been developed by Xu in [34]; much of our motivation for considering quasi-hamiltonian spaces comes from his work. Our results indicate that Dirac structures provide a natural framework for the common description of various notions of momentum maps (as e.g. in [1,22], see also [31]).We illustrate our results on multiplicative 2-forms and Dirac structures in many examples. In the case of action groupoids, we obtain an explicit formula for the natural map from the cohomology of the Cartan model of an H-manifold [2, 16] to (Borel) equivariant cohomology [2,3] in degree three; for the monodromy groupoid of a foliation F, we show that multiplicative 2-forms are closely related to the usual cohomology and spectral sequence of F [19].The entire discussion of relatively closed multiplicative 2-forms on groupoids may be embedded in the more general context of a van Est theorem for the "bar-de Rham" double complex of forms on the simplicial space of composable sequences in a groupoid G, whose total complex computes the cohomology of the classifying space...

We discuss the integrability of Jacobi manifolds by contact groupoids, and then look at what the Jacobi point of view brings new into Poisson geometry. In particular, using contact groupoids, we prove a Kostant-type theorem on the prequantization of symplectic groupoids, which answers a question posed by Weinstein and Xu [20]. The methods used are those of Crainic-Fernandes on A-paths and monodromy group(oid)s of algebroids. In particular, most of the results we obtain are valid also in the non-integrable case.1 recall that a form ω on a Lie groupoid Σ is called multiplicative if m * ω = pr * 1 ω + pr * 2 ω, where pr 1 , pr 2 are the projections, and m is the multiplication, all defined on the space Σ 2 of pairs of composable arrows of Σ.

Abstract. C * -algebras form a 2-category with * -homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions, and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby-Smith twisted actions and equivalence of such actions, covariant representations, and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green and Busby-Smith.The Packer-Raeburn Stabilisation Trick implies that all Busby-Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalise this to actions of strict 2-groupoids.

Lie algebroids cannot always be integrated into Lie groupoids. We introduce a new structure, 'Weinstein groupoid', which may be viewed as stacky groupoids. We use this structure to present a solution to the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie algebroid can be integrated into such a groupoid.

In this paper, we study Lie 2-bialgebras, paying special attention to coboundary ones, with the help of the cohomology theory of L∞-algebras with coefficients in L∞-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras (also known as pre-Lie algebras) and symplectic Lie algebras (also called quasi-Frobenius Lie algebras). * Research supported by NSFC (10920161, 11101179, 11271202), SRFDP (200800550015, 20100061120096) and the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen. 0 Keyword: L∞-bialgebras, Lie 2-algebras, Lie 2-bialgebras, left-symmetric algebras, symplectic Lie algebras 0 MSC: Primary 17B65. Secondary 18B40, 58H05. 1 does one allow homotopy in a Lie bialgebra structure? A very nice method is given by Kravchenko in [20] via higher derived brackets [1, 34] and Kosmann-Schwarzbach's big bracket [19]. Given a vector space V , we view the bracket l ∈ ∧ 2Then a Lie bialgebra structure on V is equivalent to l + c, l + c = 0, where ·, · is the big bracket defined by extending the natural pairing between V and V * via the graded Leibniz rule:Using this idea, Kravchenko then generalizes the above to a Z-graded vector space V • and defines an L ∞ -bialgebra. From an operadic point of view, the minimal resolution defines a homotopy-version, P ∞ , of an algebra P , if the corresponding operad (or dioperad or PROP) O P of P is Koszul. Lie bialgebras can correspond either to a PROP or to a dioperad, and both of them are proved to be Koszul in [17,33]. Taking the minimal resolution gives us a notion of an L ∞ -bialgebra which is exactly the one defined by Kravchenko above. However, in this setting, although a 2-term L ∞ -bialgebra gives a Lie 2-algebra structure on V , it does not give a Lie 2-algebra structure on the dual, V * . If one expects that a good categorification of Lie bialgebras should consist of Lie 2-algebra structures on V and V * , along with some compatibility conditions between them, then Kravchenko's L ∞ -bialgebra needs to be modified. In [13], the authors applied a simple shift to solve this problem. From an operadic point of view, such shifts have already appeared in [25] motivated from an apparently different motivation in deformation quantization.We adapt the shifting trick to our setting and give the definition of a Lie 2-bialgebra (see Definition 2.5). We observe that a first example of a Lie 2-bialgebra is a Lie bialgebra viewed as a Lie 2-bialgebra (see Remark 3.11). Furthermore, in the strict case, we describe the compatibility conditions between brackets and cobrackets as a cocycle condition (Theorem 2.10). For this, we develop the cohomology theory of an L ∞ -algebra L with coefficients in representations on k-term complexes of vector spaces (known as L ∞ -modules). When we restrict to the adjoint representation, we recover the cohomology studied in [26]. We give the adjoint representation of L in terms of the big bracket. We also introduce Manin triples in this general ...