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...

This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how construct a function associated germ any local symplectic groupoid provide an explicit (smooth, non-formal) universal formula $$S_\pi $$ integrating structure $$\pi on coordinate space. The second result involves semiclassic...

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 purpose of this Note is to unify quantum groups and star-products under a general umbrella: quantum groupoids. It is shown that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. The converse question, i.e., the quantization problem, is posed. In particular, any regular triangular Lie bialgebroid is shown quantizable. For the Lie bialgebroid of a Poisson mani...