In this article we discuss some general results on the covariant Picard groupoid in the context of differential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.

The coordinate projective line over a field is seen as a groupoid with a further ‘projection’ structure. We investigate conversely to what extent such an, abstractly given, groupoid may be coordinatized by a suitable field constructed out of the geometry.

We consider the union of two pants decompositions of the same orientable 2-dimensional surface of any genus g. Each pants decomposition corresponds to a handlebody bounded by this surface, so two pants decompositions correspond to a Heegaard splitting of a 3manifold. We introduce a groupoid acting on double pants decompositions. This groupoid is generated by two simple transformations (called f...

Multiple twinning in cubic crystals is represented geometrically by a three-dimensional fractal and algebraically by a groupoid. In this groupoid, the variant crystals are the objects, the misorientations between the variants are the operations, and the Sigma3(n) operators are the different types of operations (expressed by sets of equivalent operations). A general formula gives the number of v...

We construct an endofunctor of paths in the category of small category and show how to construct the standard homotopy invariants from it. We give a novel proof that the fundamental groupoid of a category is its associated universal groupoid.

In 1949, M. Sholander [4] showed that every medial cancellation groupoid can be embedded into a medial quasigroup. In this note we prove the dual assertion, that every medial division groupoid is a homomorphic image of a medial quasigroup.

An AG-groupoid is a non-associative algebraic structure mid way between a groupoid and a commutative semigroup. The left identity in an AGgroupoid if exists is unique [9]. An AG-groupoid is non-associative and non-commutative algebraic structure, nevertheless, it posses many interesting properties which we usually find in associative and commutative algebraic structures. An AG-groupoid with rig...

The twisted Drinfeld double (or quasi-quantum double) of a finite group with a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3-cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties ...