We define the thin fundamental Gray 3-groupoid S3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M) → C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.

In this paper, we review the application of symmetry groupoids in coupled cell networks, which intuitively speaking allows for the modeling of local symmetry and synchrony in directed graphs. We draw directly and heavily from the works of Golubitsky and Stewart (see [5], [11]) while presenting the groupoid-theoretic approach, in addition to providing a few original examples to motivate the theo...

Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general Lie-Rinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known as triples) and the associated standard constructions. This extends a characterization of equivari...

Given an ample groupoid, we construct a spectral sequence with groupoid homology integer coefficients on the second sheet, converging to K-groups of (reduced) C*-algebra, provided has torsion-free stabilizers and satisfies strong form Baum-Connes conjecture. The construction is based triangulated category approach conjecture developed by Meyer Nest. We also present few applications topological ...

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...