A van Kampen theorem for equivariant fundamental groupoids

The equivariant fundamental groupoid of a GG-space XX is a category which generalizes the fundamental groupoid of a space to the equivariant setting. In this paper, we prove a van Kampen theorem for these categories: the equivariant fundamental groupoid of XX can be obtained as a pushout of the categories associated to two open GG-subsets covering XX. This is proved by interpreting the equivariant fundamental groupoid as a Grothendieck semidirect product construction, and combining general properties of this construction with the ordinary (non-equivariant) van Kampen theorem. We then illustrate applications of this theorem by showing that the equivariant fundamental groupoid of a GG-CW complex only depends on the 2-skeleton and also by using the theorem to compute an example.