We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We extend the construction of the tangent prolongation of an abstract Lie algebroid to show that the Lie algebroid structure of any LA-groupoid may be prolonged to the Lie algebroid of its groupoid structure. In the case of a double groupoid, this prolonged structure for either LA-groupoid is canonically isomorphic to the Lie algebroid structure associated with the other; this extends many canonical isomorphisms associated with iterated tangent and cotangent structures. We calculate several examples from Poisson geometry. We show that the cotangent of any double Lie groupoid is a symplectic double groupoid and that the side groupoids of a symplectic double groupoid are Poisson groupoids in duality; thus the duals of the LA-groupoids of any double groupoid are a pair of Poisson groupoids in duality.

theory, and change of base for groupoids and multiple

We prove that under certain mild assumptions a Lie bialgebroid integrates to a Poisson groupoid. This includes, in particular, a new proof of the existence of local symplectic groupoids for any Poisson manifold, a theorem of Karasev and of Weinstein.

Abstract. As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2-cells of the horizontal 2-category. As a special case, strict 2-algebras with one object and everything invertible are crossed modules under a group.

We outline the main features of the definitions and applications of crossed complexes and cubical #-groupoids with connections.

We show that the Manin triple characterization of Lie bialgebras in terms of the Drinfel'd double may be extended to arbitrary Poisson manifolds and indeed Lie bialgebroids by using double cotangent bundles, rather than the direct sum structures (Courant algebroids) utilized for similar purposes by Liu, Weinstein and Xu. This is achieved in terms of an abstract notion of double Lie algebroid (where "double" is now used in the Ehresmann sense) which unifies many iterated constructions in differential geometry.

All spaces are assumed to be locally path connected and semi-locally 1-connected. Let X be a connected topological group with identity e, and let p: ˜ X → X be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any point ˜e in ˜ X with p˜e = e there is a unique structure of topological group on ˜ X such that ˜e is the

Abstract. The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal grouplike element and quantum and Frobenius-Perron dimensions of simple objects are computed. Contents

This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids.

Abstract. Associated with the canonical symplectic structure on a cotangent bundle T ∗ M is the diffeomorphism #: T ∗ (T ∗ M) − → T(T ∗ M). This and the Tulczyjew diffeomorphism T(T ∗ M) − → T ∗ (TM) may be derived from the canonical involution T(TM) − → T(TM) by suitable dualizations. We show that the constructions which yield these maps extend very generally to the double Lie algebroids of double Lie groupoids, where they play a crucial role in the relations between double Lie algebroids and Lie bialgebroids. There have been several talks this meeting about notions of double for Lie bialgebroids. Some of these have derived from the 1997 construction of Liu Zhang– Ju, Alan Weinstein and Xu Ping [4] in which they introduced the notion of Courant algebroid, and some have involved elements of super mathematics. I very much hope that before long there will be a clear account of the relations between these various approaches and even a unification of them. There is another approach to the question of doubles, which was not at first related to Lie bialgebroids, but arose out of broad considerations of what may be called “second–order geometry”. It is not possible to describe this approach from scratch in an hour, but it is appropriate at this conference to indicate the broad features of it. This talk therefore takes a slice through the papers [5], [7], [6], [8], transverse to their chronological sequence and provides an alternative route to approach them. Some aspects of §2 and §4 are new. I am very grateful to Ted Voronov and Mike Prest for the splendid opportunities and good fellowship which the Workshop provided, and to the London Mathematical Society for its support. I also wish to thank Yvette Kosmann–Schwarzbach for her comments on an earlier version. 1. The three canonical diffeomorphisms with which the paper begins are associated with iterated tangent and cotangent bundles: (1) The canonical involution on an iterated tangent bundle