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...

We present a number of findings concerning groupoid dynamical systems and groupoid crossed products. The primary result is an identification of the spectrum of the groupoid crossed product when the groupoid has continuously varying abelian stabilizers and a well behaved orbit space. In this case, the spectrum of the crossed product is homeomorphic, via an induction map, to a quotient of the spe...

The notion of a fuzzy upper bound over a groupoid is introduced and some properties of it are investigated. We also define the notions of an either-or subset of a groupoid and a strong either-or subset of a groupoid and study some of their related properties. In particular, we consider fuzzy upper bounds in Bin(X), where Bin(X) is the collection of all groupoids. Finally, we define a fuzzy-d-su...

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...