Equivariant covering spaces and homotopy covering spaces

Abstract

Nonequivariantly, covering spaces over a connected (locally nice) space $X$ are in one-to-one correspondence with actions of the fundamental group of $X$ on discrete sets. For nonconnected spaces we consider instead actions of the fundamental groupoid. In this paper we generalize to the equivariant case, showing that we can use either of two possible notions of action of the equivariant fundamental groupoid. We consider both equivariant covering spaces and the more general notion of equivariant homotopy covering spaces.