Gradings, smash products and Galois coverings of a small category

TL;DR

This paper develops a theory of coverings for small categories, establishing equivalences with fundamental groupoids and constructing Galois coverings via smash products, including the universal cover.

Contribution

It introduces effective gradings and explicit constructions of Galois coverings for small categories, extending concepts from linear cases.

Findings

Category of Galois coverings is equivalent to that of fundamental groupoid coverings

Universal cover can be obtained from the fundamental groupoid

Explicit construction of Galois coverings using smash products

Abstract

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.