Van Kampen’s theorem for fundamental groups may be stated as follows:

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 $\pi_{1}(X,A)$ on a set $A$ of base points, [1]. This groupoid consists of homotopy classes rel end points of paths in $X$ joining points of $A\cap X$. In particular, if $X$ is a contractible space, and $A$ consists of two distinct points of $X$, then $\pi_{1}(X,A)$ is easily seen to be isomorphic to the groupoid often written $\mathcal{I}$ 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.

The interpretation of this theorem as a calculational tool for fundamental groups needs some development of ‘combinatorial groupoid theory’, [2, 4]. This theorem implies the calculation of the fundamental group of the circle as the group of integers, since the group of integers is obtained from the groupoid $\mathcal{I}$ by identifying, in the category of groupoids, its two vertices.

There is a version of the last theorem when $X$ is covered by the union of the interiors of a family $\{U_{\lambda}:\lambda\in\Lambda\}$ of subsets, [3]. The conclusion is that if $A$ meets each path component of all 1,2,3-fold intersections of the sets $U_{\lambda}$, then A meets all path components of $X$ and the diagram

of morphisms induced by inclusions is a coequaliser in the category of groupoids.