For a fixed quasigroup Q, equivalences are established between the following categories: (i) the category of modules over the quasigroup; (ii) the category of representations of a stabilizer in the universal multiplication group; (iii) the category of representations of the fundamental groupoid of the Cayley diagram of the quasigroup in the category of abelian groups. A generalization to the right quasigroup case is also described. The equivalences are specialized to varieties, in particular the variety of Quandles.