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...

We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pull-back or push-forward formulas for integrals over suitable groupoids.

Tamsamani’s weak n-groupoids are known to model n-types. In this paper we show that every Tamsamani weak n-groupoid representing a connected n-type is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak n-groupoids and catn−1-groups as models of connected n-types.

We adapt the generalization of root systems of the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields that there exist infinitely many “standard”, but only 9 “except...

The present paper is a comprehensive survey of non-indempotent left distributive left quasigroups. It contains several new results about free groupoids and normal forms of terms in certain subvarieties. It is a continuation of a series of papers on selfdistributive groupoids, started by [KepN,03].

Homotopy 3-types can be modelled algebraically by Tamsamani’s weak 3-groupoids as well as, in the path connected case, by cat-groups. This paper gives a comparison between the two models in the path-connected case. This leads to two different semistrict algebraic models of connected 3-types using Tamsamani’s model. Both are then related to Gray groupoids.

The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache f, g principal isotopes and G-Smarandache loops.

Symmetry groups or groupoids of C∗-algebras associated to nonHausdorff spaces are often non-Hausdorff as well. We describe such symmetries using crossed modules of groupoids. We define actions of crossed modules on C∗-algebras and crossed products for such actions, and justify these definitions with some basic general results and examples.

Poisson manifolds may be regarded as the infinitesimal form of symplectic groupoids. Twisted Poisson manifolds considered by Ševera and Weinstein [14] are a natural generalization of the former which also arises in string theory. In this note it is proved that twisted Poisson manifolds are in bijection with a (possibly singular) twisted version of symplectic groupoids.

We achieve a classification of n-types of simplicial presheaves in terms of (n− 1)-types of presheaves of simplicial groupoids. This can be viewed as a description of the homotopy theory of higher stacks. As a special case we obtain a good homotopy theory of (weak) higher groupoids.