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 pro-longation 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 ei...

The notion of H-covariant strong Morita equivalence is introduced for ∗-algebras over C = R(i) with an ordered ring R which are equipped with a ∗-action of a Hopf ∗-algebra H . This defines a corresponding H-covariant strong Picard groupoid which encodes the entire Morita theory. Dropping the positivity conditions one obtains H-covariant ∗-Morita equivalence with its H-covariant ∗-Picard groupo...

‎In this paper‎, ‎we introduce the structure of a groupoid associated to a vector field‎ ‎on a smooth manifold‎. ‎We show that in the case of the $1$-dimensional manifolds‎, ‎our‎ ‎groupoid has a‎ ‎smooth structure such that makes it into a Lie groupoid‎. ‎Using this approach‎, ‎we associated to‎ ‎every vector field an equivalence‎ ‎relation on the Lie algebra of all vector fields on the smooth...

We approach Mackenzie’s LA-groupoids from a supergeometric point of view by introducing Q-groupoids. A Q-groupoid is a groupoid object in the category of Q-manifolds, and there is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. Using this approach, we associate to every LA-groupoid a double complex whose cohomology simultaneously generalizes Lie groupoid coh...

The notion of travel groupoids was introduced by L. Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set V and a binary operation ∗ on V satisfying two axioms. For a travel groupoid, we can associate a graph. We say that a graph G has a travel groupoid if the graph associated with the travel groupoid is equal to G. Nebeský gave a characterization f...

A symplectic groupoid G. := (G1 ⇉ G0) determines a Poisson structure on G0. In this case, we call G. a symplectic groupoid of the Poisson manifold G0. However, not every Poisson manifold M has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing Morita equivalence in the category of all Poisson manifolds. In this paper, we construct symplectic Wein...

In this note a functorial approach to the integration problem of an LAgroupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibered products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns out, that the fibered product of Lie algebroids along integrable morphisms is...

The line digraph of the Cayley color graph of a transitive groupoid can be colored so that the groupoid of partial automorphisms is isomorphic to a semidirect product of the original groupoid.

R esum e: Nous d emontrons que, sous des circomstances g en erales, l'union disjoint des couvertes universales des etoiles d'un groupo de de Lie admet le structure d'un groupo de de Lie auquel que le projection a une propri et e de monodromie sur les extensions des morphismes emousse. Ca compl etes une conte d etailles des r esultats annonc es par J. Pradines. Introduction The notion of monodro...