We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying the identities x(yz) ≈ (xy)(xz) and (xx)y ≈ xy) modulo the lattice of subvarieties of left distributive idempotent groupoids. A free groupoid in a subvariety of LDLI groupoids satisfying an identity x ≈ x decomposes as the direct product of its largest idempotent factor and a ...

In this paper we introduced the notions of uni-soft AG-groupoids, uni-soft left(right) ideals and uni-soft quasi ideals of AG-groupoids and also investigated related properties of these structures. Some interesting Characterizations of uni-soft AG-groupoids and uni-soft left(right) ideals have been explored. Mathematics Subject Classification: 42B20, 42B25, 46E30

We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allows us to establish a linearization theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of...

Work of Jean Renault shows that, for topologically principal étale groupoids, a diagonal-preserving isomorphism of reduced C ∗-algebras yields an isomorphism of groupoids. Several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses. In this paper, we extend the known results by allowing more general coefficient rings and by we...

This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...

The goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps need to be thought of as a 2-category. This 2-category may be either taken to be the weak 2-category of groupoids, bibundles and equivariant maps between bibundles or the strict 2category of geometric stacks represented by proper etale Lie groupoids. While nothing in this p...

2 Ep-Groupoids and Generalized Maps 17 2.1 Ep-Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Functors and Equivalences . . . . . . . . . . . . . . . . . . . . 21 2.3 Inversion of Equivalences and Generalized Maps . . . . . . . . 28 2.4 Strong Bundles over Ep-Groupoids . . . . . . . . . . . . . . . 33 2.5 Auxiliary Norms . . . . . . . . . . . . . . . . . . . . . . . . . 4...