The first five sections of this paper are a survey of algebras of pseudodifferential operators on groupoids. We thus review differentiable groupoids, the definition of pseudodifferential operators on groupoids, and some of their properties. We use then this background material to establish a few new results on these algebras that are useful for the analysis of geometric operators on non-compact...

We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, whic...

By a paramedial groupoid we mean a groupoid satisfying the equation xy · zu = uy · zx. As it is easy to see, the equational theory of paramedial groupoids, as well as the equational theory based on any balanced equation, is decidable. In this paper we are going to investigate the equational theory of paramedial cancellation groupoids; by this we mean the set of all equations satisfied by parame...

In 1968, when this book was written, categories had been around for 20 years and groupoids for twice as long. Category theory had by then become widely accepted as an essential tool in many parts of mathematics and a number of books on the subject had appeared, or were about to appear (e.g. [13, 22, 37, 58, 65]). By contrast, the use of groupoids was confined to a small number of pioneering art...

In this paper, we introduce fuzzy rank functions for groupoids, and we investigate their roles in the semigroup of binary systems by using the notions of right parallelisms and [Formula: see text]-shrinking groupoids.

We establish a functor Kan from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids.

This is an introductory survey of the passage from groups to groupoids and their higher dimensional versions, with most emphasis on calculations with crossed modules and the construction and use of homotopy double groupoids.

We fill a lacuna in the literature by giving a version in dimension 1 of the Relative Hurewicz Theorem, and relate this to abelianisations of groupoids, covering spaces, covering morphisms of groupoids, and Crowell’s notion of derived modules.

This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored.

This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored. 1