nLab on fundamental infinity-groupoid
nlab on covering space
Goerss-Jardine, section III.1. It is the left adjoint to a classifying space functor. The fundamental groupoid of a classifying space of a small cat is the free groupoid on that cat. A consequence is that the fundamental group of the nerve of the Q-construction of an exact cat is isomorphic to the 0th K-group of the cat.
nLab page on Fundamental groupoid