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 ...

Tarski associative groupoid (TA-groupoid) is a kind of non-associative satisfying law. In this paper, the new notions transposition regular TA-groupoid are proposed and their properties structural characteristics studied by using band quasi-separativity. particular, following conclusions strictly proved: (1) every left semigroup; (2) disjoint union sub Abelian groups; (3) finite with quasi-sepa...

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These rep...

In ([L2]), Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In an annex of [E6], Lesieur’s axioms have been simplified. In this article, we suppose that the basis is central; in that case, we prove that a specific sub-C∗ algebra is, in a sense, invariant under all the data which define the measured quantum group, which allow us to prov...

Let f : G → A be a surjective homomorphism of transitive groupoid schemes and let L denote the kernel of f . The exact sequence of groupoid schemes 1 → L → G → A → 1 induces a sequence of functors between the categories of finite representations of these groupoid schemes Repf (A) → Repf (G) → Repf (L). We show that the category Repf (L) is a quotient category of Repf (G) by Repf (A) in an appro...

In this paper, we have introduced the concept of fuzzy ordered AG-groupoids which is the generalization of fuzzy ordered semigroups first considered by Kehayopulu and Tsingelis (2002). We have studied some important features of a left regular ordered AG-groupoid in terms of fuzzy left ideals, fuzzy right ideals, fuzzy two-sided ideals, fuzzy generalized bi-ideals, fuzzy bi-ideals, fuzzy interio...