We show that the graded set of filter homotopy classes rel vertices of maps from the n-globe to a filtered space may be given the structure of (strict) globular ω-groupoid. The proofs use an analogous fundamental cubical ω-groupoid due to the author and Philip Higgins in 1981. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation all...

We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in th...

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C-algebra may be regarded as a result of a quantization procedure. The C-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra G) is the C-algebra of a continuous field of C-algebras over R with fibers ...

Suppose G is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let G/G denote the orbit space of G and C∗(G) denote the groupoid C∗-algebra. Suppose that the isotropy groups of G are amenable. We show that C∗(G) is CCR if and only if G/G is a T1 topological space and all of the isotropy groups are CCR. We also show that C∗(G) is GCR if and only if G/G is a T...

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...