Gradings, smash products and Galois coverings of a small category

In this paper we develop the theory of coverings of a small connected category B. We show that the category of Galois coverings of B is equivalent to the category of Galois coverings of its fundamental groupoid. Making use of effective gradings of B we explicitly construct Galois coverings through a smash product analogous to the one considered in the linear case. In particular, the universal cover of B can be obtained from its fundamental groupoid.