The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.

In this paper, we introduce the notion of a logistic groupoid on the real numbers R, and show that, given a groupoid (R, ?) with some conditions, there exists a groupoid (X, ~) such that (R, ?) is the logistic groupoid of (X, ~).

We define a notion of connection in fibre bundle that is compatible with singular foliation the base. Fibre bundles equipped such connections are plentiful supply, arising naturally for any Lie groupoid-equivariant bundle, and simultaneously generalising regularly foliated sense Kamber-Tondeur foliations. hierarchies diffeological holonomy groupoids associated to bundles, which arise from paral...

The main result of this paper is a convexity theorem for momentum mappings J :M → g∗ of certain hamiltonian actions of noncompact semisimple Lie groups. The image of J is required to fall within a certain open subset D of g∗ which corresponds roughly via the orbit method to the discrete series of representations of the group G. In addition, J is required to be proper as a map from M to D. A rel...

The main result of this paper is a convexity theorem for momentum mappings J :M → g∗ of certain hamiltonian actions of noncompact semisimple Lie groups. The image of J is required to fall within a certain open subset D of g∗ which corresponds roughly via the orbit method to the discrete series of representations of the group G. In addition, J is required to be proper as a map from M to D. A rel...

Abstract We give a new construction of the holonomy and fundamental groupoids singular foliation. In contrast with existing Androulidakis Skandalis, our method proceeds by taking quotient an infinite-dimensional space paths. This strategy is direct extension classical for regular foliations mirrors integration Lie algebroids via paths (per Crainic Fernandes). this way, we obtain characterizatio...

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a...

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...

Abstract. We build a concrete and natural model for the strict 2-category of orbifolds. In particular we prove that if one localizes the 2-category of proper etale Lie groupoids at a class of 1-arrows that we call “covers,” then the strict 2-category structure drops down to the localization. In our construction the spaces of 1and 2-arrows admit natural topologies, the space of morphisms (1-arro...