This unpublished note contains some materials taken from my old study note on groupoids and small categories ([1]). It contains a proof for the fact that a groupoid is any group bundle over an equivalence relation. Moreover, the action of a category G on a category H as well as the resulting semi-direct product category H ×α G will be defined (when either G is a groupoid or H (0) = G). If both ...

We refine the concept of the symmetry group of a geometric object through its symmetry groupoid, which incorporates both global and local symmetries in a common framework. The symmetry groupoid is related to the weighted differential invariant signature of a submanifold, that is introduced to capture its fine grain equivalence and symmetry properties. Applications to the recognition and symmetr...

We associate a canonical unital involutive quantale to a topological groupoid. When the groupoid is also étale, this association is compatible with but independent from the theory of localic étale groupoids and their quantales [19] of P. Resende. As a motivating example, we describe the connection between the quantale and the C∗-algebra that both classify Penrose tilings, which was left as an o...

We generalise the Margolis-Meakin graph expansion of a group to a construction for ordered groupoids, and show that the graph expansion of an ordered groupoid enjoys structural properties analogous to those for graph expansions of groups. We also use the Cayley graph of an ordered groupoid to prove a version of McAlister’s P–theorem for incompressible ordered groupoids. AMS 2000 Mathematics sub...

It is well known that a measured groupoid G defines a von Neu-mann algebra W * (G), and that a Lie groupoid G canonically defines both a C *-algebra C * (G) and a Poisson manifold A * (G). We show that the maps (G) are functorial with respect to suitable categories. In these categories Morita equivalence is isomorphism of objects, so that these maps preserve Morita equivalence.

k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of CuntzKrieger type. Here we develop a theory of the fundamental groupoid of a kgraph, and relate it to the fundamental groupoid of an associated graph called the 1-skeleton. We also explore the failure, in general, of k-graphs to faithfully embed into their fu...

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...