A global action is the algebraic analogue of a topological manifold. This construction was introduced in first place by A. Bak as a combinatorial approach to K-Theory and the concept was later generalized by Bak, Brown, Minian and Porter to the notion of groupoid atlas. In this paper we define and investigate homotopy invariants of global actions and groupoid atlases, such as the strong fundame...

The groupoid interpretation of Martin-Löf type theory not only shows the independence of uniqueness of identity proofs from the axioms of intensional type theory but is also constructive and validates the computation rules as definitional equalities. The groupoid semantics are very clear when interpreting dependent types and in particular the identity types but less so when defining equality pr...

In this work we prove that the set of congruences on an nd-groupoid under suitable conditions is a complete lattice which is a sublattice of the lattice of equivalence relations on the nd-groupoid. The study of these conditions allowed to construct a counterexample to the statement that the set of (fuzzy) congruences on a hypergroupoid is a complete lattice.

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold group-oid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an action of a bundle of compact topological...

In this paper, we apply the theory of inverse semigroups to the C∗-algebra U [Z] considered in [Cun08]. We show that the C∗-algebra U [Z] is generated by an inverse semigroup of partial isometries. We explicity identify the groupoid Gtight associated to the inverse semigroup and show that Gtight is exactly the same groupoid obtained in [CL10].

This is a sequel to the paper [6] ‘Homology of types in model theory I: Basic concepts and connections with type amalgamation’. We compute the group H2 for strong types in stable theories and show that any profinite abelian group can occur as the group H2 in the model-theoretic context. The work described in this paper was originally inspired by Hrushovski’s discovery [8] of striking connection...

in this paper we construct the category of coverings of fundamental generalized lie group-groupoid associatedwith a connected generalized lie group. we show that this category is equivalent to the category of coverings of aconnected generalized lie group. in addition, we prove the category of coverings of generalized lie groupgroupoidand the category of actions of this generalized lie group-gro...

‎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 introduce a new kind of groupoid—a pseudo étale groupoid, which provides many interesting examples of noncommutative Poisson algebras as defined by Block, Getzler, and Xu. Following the idea that symplectic and Poisson geometries are the semiclassical limits of the corresponding quantum geometries, we quantize these noncommutative Poisson manifolds in the framework of deformation quantizatio...