On the relation between the fundamental groupoids of the classifying space and the nerve of an open cover (Q1069175)

Associated to an open cover U of a topological space X are the classical simplicial nerve of U, KU, and the classifying space of U, BU. There is a natural map BU\(\to KU\) and an induced morphism on fundamental groupoids \(\pi\). In the case that all the elements of the cover are open and path connected the authors show that \(\pi\) BU\(\to \pi KU\) is a quotient morphism and compute the normal subgroupoid in terms of generators and relations. Several relationships are noted between the nature of the morphism \(\pi\) BU\(\to \pi KU\) and work of others concerning actions of groups and transformation groups.