Van Kampen’s theorem
Van Kampen’s theorem for fundamental groups may be stated as follows:
Let be a topological space which is the union of the interiors of two path connected subspaces . Suppose is path connected. Let further and , be induced by the inclusions for . Then is path connected and the natural morphism
Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups.
The notion of pushout in the category of groupoids allows for a version of the theorem for the non path connected case, using the fundamental groupoid on a set of base points, [rb1]. This groupoid consists of homotopy classes rel end points of paths in joining points of . In particular, if is a contractible space, and consists of two distinct points of , then is easily seen to be isomorphic to the groupoid often written with two vertices and exactly one morphism between any two vertices. This groupoid plays a role in the theory of groupoids analogous to that of the group of integers in the theory of groups.