We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step towards a general continuous cohomology theory for topological racks and quandles.

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their crossproducts, and introduce representation theory of racks and quandles. We prove several results on the strong representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the Pontryagin dual of a rack as an Abelian group which, in the finite involutive connected case, coincides with the set of its strong irreducible representations.

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International audienceWe define a group BrR(G) containing, in a sense, the graded complex and orthogonal Brauer groups of a locally compact groupoid G equipped with an involution. When the involution is trivial, we show that the new group naturally provides a generalisation of Donovan-Karoubi's graded orthogonal Brauer group GBrO. More generally, it is shown to be a direct summand of the well-known graded complex Brauer goup. In addition, we prove that BrR(G) identifies with a direct sum of a Real cohomology group and the abelian group ExtR(G, S 1) of Real graded S 1-central extensions. A cohomological picture is then given

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