Covering Space - Relations With Groupoids

Relations With Groupoids

One of the ways of expressing the algebraic content of the theory of covering spaces is using groupoids and the fundamental groupoid. The latter functor gives an equivalence of categories

between the category of covering spaces of a reasonably nice space X and the category of groupoid covering morphisms of π₁(X). Thus a particular kind of map of spaces is well modelled by a particular kind of morphism of groupoids. The category of covering morphisms of a groupoid G is also equivalent to the category of actions of G on sets, and this allows the recovery of more traditional classifications of coverings. Proofs of these facts are given in the book 'Topology and Groupoids' referenced below.